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Properties

Label 2-171-19.7-c3-0-22
Degree 22
Conductor 171171
Sign −0.288−0.957i-0.288 - 0.957i
Analytic cond. 10.089310.0893
Root an. cond. 3.176373.17637
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  + (−1.45 − 2.52i)2-s + (−0.241 + 0.419i)4-s + (−10.1 − 17.4i)5-s − 1.64·7-s − 21.8·8-s + (−29.4 + 50.9i)10-s + 8.92·11-s + (19.6 − 34.0i)13-s + (2.39 + 4.14i)14-s + (33.8 + 58.5i)16-s + (−3.95 − 6.84i)17-s + (−42.6 + 70.9i)19-s + 9.77·20-s + (−12.9 − 22.5i)22-s + (55.2 − 95.7i)23-s + ⋯
L(s)  = 1  + (−0.514 − 0.891i)2-s + (−0.0302 + 0.0523i)4-s + (−0.903 − 1.56i)5-s − 0.0887·7-s − 0.967·8-s + (−0.930 + 1.61i)10-s + 0.244·11-s + (0.419 − 0.726i)13-s + (0.0456 + 0.0791i)14-s + (0.528 + 0.915i)16-s + (−0.0563 − 0.0976i)17-s + (−0.515 + 0.856i)19-s + 0.109·20-s + (−0.125 − 0.218i)22-s + (0.501 − 0.868i)23-s + ⋯

Functional equation

Λ(s)=(171s/2 ΓC(s) L(s)=((−0.288−0.957i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(171s/2 ΓC(s+3/2) L(s)=((−0.288−0.957i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 171171    =    32⋅193^{2} \cdot 19
Sign: −0.288−0.957i-0.288 - 0.957i
Analytic conductor: 10.089310.0893
Root analytic conductor: 3.176373.17637
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ171(64,⋅)\chi_{171} (64, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 171, ( :3/2), −0.288−0.957i)(2,\ 171,\ (\ :3/2),\ -0.288 - 0.957i)

Particular Values

L(2)L(2) ≈\approx 0.298499+0.401795i0.298499 + 0.401795i
L(12)L(\frac12) ≈\approx 0.298499+0.401795i0.298499 + 0.401795i
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)
bad3 1 1
19 1+(42.6−70.9i)T 1 + (42.6 - 70.9i)T
good2 1+(1.45+2.52i)T+(−4+6.92i)T2 1 + (1.45 + 2.52i)T + (-4 + 6.92i)T^{2}
5 1+(10.1+17.4i)T+(−62.5+108.i)T2 1 + (10.1 + 17.4i)T + (-62.5 + 108. i)T^{2}
7 1+1.64T+343T2 1 + 1.64T + 343T^{2}
11 1−8.92T+1.33e3T2 1 - 8.92T + 1.33e3T^{2}
13 1+(−19.6+34.0i)T+(−1.09e3−1.90e3i)T2 1 + (-19.6 + 34.0i)T + (-1.09e3 - 1.90e3i)T^{2}
17 1+(3.95+6.84i)T+(−2.45e3+4.25e3i)T2 1 + (3.95 + 6.84i)T + (-2.45e3 + 4.25e3i)T^{2}
23 1+(−55.2+95.7i)T+(−6.08e3−1.05e4i)T2 1 + (-55.2 + 95.7i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(71.8−124.i)T+(−1.21e4−2.11e4i)T2 1 + (71.8 - 124. i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+57.1T+2.97e4T2 1 + 57.1T + 2.97e4T^{2}
37 1−297.T+5.06e4T2 1 - 297.T + 5.06e4T^{2}
41 1+(−112.−194.i)T+(−3.44e4+5.96e4i)T2 1 + (-112. - 194. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(163.+283.i)T+(−3.97e4+6.88e4i)T2 1 + (163. + 283. i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1+(56.2−97.4i)T+(−5.19e4−8.99e4i)T2 1 + (56.2 - 97.4i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(305.−529.i)T+(−7.44e4−1.28e5i)T2 1 + (305. - 529. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(368.+638.i)T+(−1.02e5+1.77e5i)T2 1 + (368. + 638. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−420.+728.i)T+(−1.13e5−1.96e5i)T2 1 + (-420. + 728. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(53.2−92.1i)T+(−1.50e5−2.60e5i)T2 1 + (53.2 - 92.1i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+(386.+669.i)T+(−1.78e5+3.09e5i)T2 1 + (386. + 669. i)T + (-1.78e5 + 3.09e5i)T^{2}
73 1+(369.+639.i)T+(−1.94e5+3.36e5i)T2 1 + (369. + 639. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(257.+446.i)T+(−2.46e5+4.26e5i)T2 1 + (257. + 446. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+618.T+5.71e5T2 1 + 618.T + 5.71e5T^{2}
89 1+(62.3−107.i)T+(−3.52e5−6.10e5i)T2 1 + (62.3 - 107. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+(349.+606.i)T+(−4.56e5+7.90e5i)T2 1 + (349. + 606. i)T + (-4.56e5 + 7.90e5i)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

−11.55137135701904150733923367774, −10.73153283698504006761161849856, −9.507259498961522294142052683874, −8.712622945735404713410287881832, −7.890643929019666148643131047081, −6.08638993667591146257228610411, −4.75602303906750676462328901969, −3.41519181344561502067669721455, −1.46665916810169550787583710691, −0.28071650346212535961259490371, 2.79511722399199797198566569524, 3.98710080779863320291732249302, 6.11495964908182613753303185922, 6.92393442532990976664271374508, 7.58986024081959069744637722648, 8.679103589521430418852986488855, 9.835930524457245862108419925327, 11.33539545878912318605609809039, 11.48584525053089852312857577670, 13.06103467785953648364178936554

Graph of the ZZ-function along the critical line