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Properties

Label 2-425-17.2-c1-0-6
Degree 22
Conductor 425425
Sign 0.914+0.403i0.914 + 0.403i
Analytic cond. 3.393643.39364
Root an. cond. 1.842181.84218
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.66 − 1.66i)2-s + (1.44 − 0.600i)3-s + 3.53i·4-s + (−3.41 − 1.41i)6-s + (−1.37 + 3.30i)7-s + (2.54 − 2.54i)8-s + (−0.379 + 0.379i)9-s + (2.29 + 0.950i)11-s + (2.12 + 5.12i)12-s − 1.25i·13-s + (7.78 − 3.22i)14-s − 1.40·16-s + (3.84 + 1.48i)17-s + 1.26·18-s + (−2.56 − 2.56i)19-s + ⋯
L(s)  = 1  + (−1.17 − 1.17i)2-s + (0.837 − 0.346i)3-s + 1.76i·4-s + (−1.39 − 0.576i)6-s + (−0.518 + 1.25i)7-s + (0.900 − 0.900i)8-s + (−0.126 + 0.126i)9-s + (0.691 + 0.286i)11-s + (0.612 + 1.47i)12-s − 0.349i·13-s + (2.08 − 0.861i)14-s − 0.352·16-s + (0.932 + 0.360i)17-s + 0.297·18-s + (−0.589 − 0.589i)19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.914+0.403i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((0.914+0.403i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.914+0.403i0.914 + 0.403i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(376,⋅)\chi_{425} (376, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), 0.914+0.403i)(2,\ 425,\ (\ :1/2),\ 0.914 + 0.403i)

Particular Values

L(1)L(1) ≈\approx 0.893433−0.188413i0.893433 - 0.188413i
L(12)L(\frac12) ≈\approx 0.893433−0.188413i0.893433 - 0.188413i
L(32)L(\frac{3}{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 1
17 1+(−3.84−1.48i)T 1 + (-3.84 - 1.48i)T
good2 1+(1.66+1.66i)T+2iT2 1 + (1.66 + 1.66i)T + 2iT^{2}
3 1+(−1.44+0.600i)T+(2.12−2.12i)T2 1 + (-1.44 + 0.600i)T + (2.12 - 2.12i)T^{2}
7 1+(1.37−3.30i)T+(−4.94−4.94i)T2 1 + (1.37 - 3.30i)T + (-4.94 - 4.94i)T^{2}
11 1+(−2.29−0.950i)T+(7.77+7.77i)T2 1 + (-2.29 - 0.950i)T + (7.77 + 7.77i)T^{2}
13 1+1.25iT−13T2 1 + 1.25iT - 13T^{2}
19 1+(2.56+2.56i)T+19iT2 1 + (2.56 + 2.56i)T + 19iT^{2}
23 1+(−8.16−3.38i)T+(16.2+16.2i)T2 1 + (-8.16 - 3.38i)T + (16.2 + 16.2i)T^{2}
29 1+(−3.35−8.09i)T+(−20.5+20.5i)T2 1 + (-3.35 - 8.09i)T + (-20.5 + 20.5i)T^{2}
31 1+(−2.25+0.935i)T+(21.9−21.9i)T2 1 + (-2.25 + 0.935i)T + (21.9 - 21.9i)T^{2}
37 1+(−4.25+1.76i)T+(26.1−26.1i)T2 1 + (-4.25 + 1.76i)T + (26.1 - 26.1i)T^{2}
41 1+(1.65−3.99i)T+(−28.9−28.9i)T2 1 + (1.65 - 3.99i)T + (-28.9 - 28.9i)T^{2}
43 1+(5.33−5.33i)T−43iT2 1 + (5.33 - 5.33i)T - 43iT^{2}
47 1+11.3iT−47T2 1 + 11.3iT - 47T^{2}
53 1+(4.02+4.02i)T+53iT2 1 + (4.02 + 4.02i)T + 53iT^{2}
59 1+(−3.16+3.16i)T−59iT2 1 + (-3.16 + 3.16i)T - 59iT^{2}
61 1+(−0.0929+0.224i)T+(−43.1−43.1i)T2 1 + (-0.0929 + 0.224i)T + (-43.1 - 43.1i)T^{2}
67 1+7.23T+67T2 1 + 7.23T + 67T^{2}
71 1+(1.69−0.703i)T+(50.2−50.2i)T2 1 + (1.69 - 0.703i)T + (50.2 - 50.2i)T^{2}
73 1+(−2.09−5.05i)T+(−51.6+51.6i)T2 1 + (-2.09 - 5.05i)T + (-51.6 + 51.6i)T^{2}
79 1+(8.34+3.45i)T+(55.8+55.8i)T2 1 + (8.34 + 3.45i)T + (55.8 + 55.8i)T^{2}
83 1+(−5.17−5.17i)T+83iT2 1 + (-5.17 - 5.17i)T + 83iT^{2}
89 1+2.19iT−89T2 1 + 2.19iT - 89T^{2}
97 1+(−3.75−9.07i)T+(−68.5+68.5i)T2 1 + (-3.75 - 9.07i)T + (-68.5 + 68.5i)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

−11.09028217720129063625538255650, −10.03130767833897657852349985210, −9.174416572592750539951638317424, −8.743944133931295307658804620567, −7.964533574597542071046475156162, −6.78724493031497368770361157820, −5.30985595314609937096798349638, −3.31149592752091270799505214140, −2.72655436480540144352652834699, −1.51004209183570363217240745130, 0.866480139129127628502774399354, 3.18761230135222390086610907548, 4.36093875960255563207370292264, 6.05917599549088667774556822948, 6.78595254042098587557565437707, 7.68546200413179985589793985742, 8.501457102004496887606162259710, 9.274965138042943824484448737178, 9.926182599221075967832918669576, 10.67589093254470085444189683948

Graph of the ZZ-function along the critical line