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Properties

Label 2-102-51.32-c4-0-20
Degree $2$
Conductor $102$
Sign $-0.994 + 0.106i$
Analytic cond. $10.5437$
Root an. cond. $3.24711$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 2i)2-s + (1.73 − 8.83i)3-s − 8i·4-s + (−7.65 − 18.4i)5-s + (−14.1 − 21.1i)6-s + (3.64 + 1.50i)7-s + (−16 − 16i)8-s + (−74.9 − 30.6i)9-s + (−52.2 − 21.6i)10-s + (25.7 + 10.6i)11-s + (−70.6 − 13.9i)12-s − 9.18i·13-s + (10.3 − 4.27i)14-s + (−176. + 35.4i)15-s − 64·16-s + (106. + 268. i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.193 − 0.981i)3-s − 0.5i·4-s + (−0.306 − 0.739i)5-s + (−0.394 − 0.587i)6-s + (0.0743 + 0.0308i)7-s + (−0.250 − 0.250i)8-s + (−0.925 − 0.378i)9-s + (−0.522 − 0.216i)10-s + (0.213 + 0.0882i)11-s + (−0.490 − 0.0965i)12-s − 0.0543i·13-s + (0.0526 − 0.0217i)14-s + (−0.784 + 0.157i)15-s − 0.250·16-s + (0.368 + 0.929i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $-0.994 + 0.106i$
Analytic conductor: \(10.5437\)
Root analytic conductor: \(3.24711\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{102} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :2),\ -0.994 + 0.106i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0991106 - 1.85324i\)
\(L(\frac12)\) \(\approx\) \(0.0991106 - 1.85324i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 2i)T \)
3 \( 1 + (-1.73 + 8.83i)T \)
17 \( 1 + (-106. - 268. i)T \)
good5 \( 1 + (7.65 + 18.4i)T + (-441. + 441. i)T^{2} \)
7 \( 1 + (-3.64 - 1.50i)T + (1.69e3 + 1.69e3i)T^{2} \)
11 \( 1 + (-25.7 - 10.6i)T + (1.03e4 + 1.03e4i)T^{2} \)
13 \( 1 + 9.18iT - 2.85e4T^{2} \)
19 \( 1 + (360. + 360. i)T + 1.30e5iT^{2} \)
23 \( 1 + (-240. - 99.6i)T + (1.97e5 + 1.97e5i)T^{2} \)
29 \( 1 + (-13.8 - 33.3i)T + (-5.00e5 + 5.00e5i)T^{2} \)
31 \( 1 + (540. + 1.30e3i)T + (-6.53e5 + 6.53e5i)T^{2} \)
37 \( 1 + (-56.2 - 135. i)T + (-1.32e6 + 1.32e6i)T^{2} \)
41 \( 1 + (-520. + 1.25e3i)T + (-1.99e6 - 1.99e6i)T^{2} \)
43 \( 1 + (-868. + 868. i)T - 3.41e6iT^{2} \)
47 \( 1 - 2.73e3T + 4.87e6T^{2} \)
53 \( 1 + (-1.55e3 + 1.55e3i)T - 7.89e6iT^{2} \)
59 \( 1 + (-2.22e3 - 2.22e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-1.13e3 - 471. i)T + (9.79e6 + 9.79e6i)T^{2} \)
67 \( 1 - 7.86e3T + 2.01e7T^{2} \)
71 \( 1 + (1.08e3 - 449. i)T + (1.79e7 - 1.79e7i)T^{2} \)
73 \( 1 + (-2.73e3 + 1.13e3i)T + (2.00e7 - 2.00e7i)T^{2} \)
79 \( 1 + (-1.48e3 + 3.58e3i)T + (-2.75e7 - 2.75e7i)T^{2} \)
83 \( 1 + (1.93e3 - 1.93e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 3.54e3T + 6.27e7T^{2} \)
97 \( 1 + (8.87e3 - 3.67e3i)T + (6.25e7 - 6.25e7i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.72338210562444824394582486009, −11.84584869646783715051423512590, −10.79687789372428864167304774415, −9.160903318718867591859790864106, −8.216880028737233941618520903992, −6.83224815186984090343442990254, −5.52438884174937078199503042946, −3.99996874543523054965887540489, −2.24302560162013263528530020295, −0.70915965367659393958210451634, 2.92825570275510934645114451867, 4.08321169615221816940583524758, 5.38904057602826201101240753407, 6.77169671785866704600603937069, 8.053296730535924457810404148772, 9.243917348160345376230293106698, 10.53029749084566345349294124011, 11.38148235305293031109619189340, 12.63216634454266960859841675203, 14.16127850462867903630594429602

Graph of the $Z$-function along the critical line