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Properties

Label 2-175-175.103-c3-0-1
Degree 22
Conductor 175175
Sign −0.777−0.628i-0.777 - 0.628i
Analytic cond. 10.325310.3253
Root an. cond. 3.213303.21330
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.70 − 0.141i)2-s + (−2.31 − 2.85i)3-s + (−0.643 − 0.0676i)4-s + (8.55 + 7.20i)5-s + (5.85 + 8.05i)6-s + (−16.6 − 8.09i)7-s + (23.1 + 3.66i)8-s + (2.80 − 13.2i)9-s + (−22.1 − 20.7i)10-s + (30.3 − 6.44i)11-s + (1.29 + 1.99i)12-s + (−54.8 − 27.9i)13-s + (43.9 + 24.2i)14-s + (0.790 − 41.0i)15-s + (−57.1 − 12.1i)16-s + (20.8 + 54.2i)17-s + ⋯
L(s)  = 1  + (−0.957 − 0.0501i)2-s + (−0.444 − 0.549i)3-s + (−0.0804 − 0.00845i)4-s + (0.764 + 0.644i)5-s + (0.398 + 0.548i)6-s + (−0.899 − 0.436i)7-s + (1.02 + 0.162i)8-s + (0.104 − 0.489i)9-s + (−0.699 − 0.655i)10-s + (0.831 − 0.176i)11-s + (0.0311 + 0.0479i)12-s + (−1.17 − 0.596i)13-s + (0.839 + 0.463i)14-s + (0.0136 − 0.706i)15-s + (−0.892 − 0.189i)16-s + (0.296 + 0.773i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.777−0.628i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((−0.777−0.628i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.777−0.628i-0.777 - 0.628i
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ175(103,⋅)\chi_{175} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), −0.777−0.628i)(2,\ 175,\ (\ :3/2),\ -0.777 - 0.628i)

Particular Values

L(2)L(2) ≈\approx 0.00673058+0.0190194i0.00673058 + 0.0190194i
L(12)L(\frac12) ≈\approx 0.00673058+0.0190194i0.00673058 + 0.0190194i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(−8.55−7.20i)T 1 + (-8.55 - 7.20i)T
7 1+(16.6+8.09i)T 1 + (16.6 + 8.09i)T
good2 1+(2.70+0.141i)T+(7.95+0.836i)T2 1 + (2.70 + 0.141i)T + (7.95 + 0.836i)T^{2}
3 1+(2.31+2.85i)T+(−5.61+26.4i)T2 1 + (2.31 + 2.85i)T + (-5.61 + 26.4i)T^{2}
11 1+(−30.3+6.44i)T+(1.21e3−541.i)T2 1 + (-30.3 + 6.44i)T + (1.21e3 - 541. i)T^{2}
13 1+(54.8+27.9i)T+(1.29e3+1.77e3i)T2 1 + (54.8 + 27.9i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(−20.8−54.2i)T+(−3.65e3+3.28e3i)T2 1 + (-20.8 - 54.2i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(12.2+116.i)T+(−6.70e3+1.42e3i)T2 1 + (12.2 + 116. i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(7.28−138.i)T+(−1.21e4−1.27e3i)T2 1 + (7.28 - 138. i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(38.9−53.6i)T+(−7.53e3−2.31e4i)T2 1 + (38.9 - 53.6i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(95.5−214.i)T+(−1.99e4−2.21e4i)T2 1 + (95.5 - 214. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(158.−103.i)T+(2.06e4−4.62e4i)T2 1 + (158. - 103. i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(184.−59.9i)T+(5.57e4−4.05e4i)T2 1 + (184. - 59.9i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(361.+361.i)T+7.95e4iT2 1 + (361. + 361. i)T + 7.95e4iT^{2}
47 1+(142.+54.7i)T+(7.71e4+6.94e4i)T2 1 + (142. + 54.7i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(139.−112.i)T+(3.09e4−1.45e5i)T2 1 + (139. - 112. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(−224.+249.i)T+(−2.14e4−2.04e5i)T2 1 + (-224. + 249. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−21.6+19.5i)T+(2.37e4−2.25e5i)T2 1 + (-21.6 + 19.5i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(152.−58.4i)T+(2.23e5−2.01e5i)T2 1 + (152. - 58.4i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−814.−591.i)T+(1.10e5+3.40e5i)T2 1 + (-814. - 591. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(662.−1.01e3i)T+(−1.58e5−3.55e5i)T2 1 + (662. - 1.01e3i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(360.+809.i)T+(−3.29e5+3.66e5i)T2 1 + (360. + 809. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(−24.2+153.i)T+(−5.43e5−1.76e5i)T2 1 + (-24.2 + 153. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(453.+503.i)T+(−7.36e4+7.01e5i)T2 1 + (453. + 503. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(145.+916.i)T+(−8.68e5+2.82e5i)T2 1 + (145. + 916. i)T + (-8.68e5 + 2.82e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.71895256818765134356799428674, −11.46974739073954839932237734567, −10.30917702547196467770617926281, −9.731014630532988009718517972023, −8.850794880053864020027016921976, −7.18346340042681824122507371054, −6.77199363139231921984281112205, −5.36663646703510766931838317236, −3.42254829422624501170088409228, −1.47829721467928351451031374250, 0.01380748917246478427193977632, 1.95497545962538546312747069467, 4.26624467567141713751358582238, 5.30350936337814507183453940207, 6.60494378340039049781243454909, 7.992962401798487147227257896207, 9.225318359218388210917431190418, 9.714976819134299218087710272177, 10.32387182807938979207963750959, 11.83127021041590355882499407270

Graph of the ZZ-function along the critical line