↑

Properties

Label 2-102-51.32-c4-0-6
Degree $2$
Conductor $102$
Sign $0.208 - 0.978i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 2i)2-s + (1.58 + 8.85i)3-s − 8i·4-s + (11.1 + 26.8i)5-s + (20.8 + 14.5i)6-s + (27.8 + 11.5i)7-s + (−16 − 16i)8-s + (−75.9 + 28.0i)9-s + (75.8 + 31.4i)10-s + (−160. − 66.6i)11-s + (70.8 − 12.6i)12-s + 253. i·13-s + (78.7 − 32.6i)14-s + (−220. + 140. i)15-s − 64·16-s + (275. + 88.3i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.175 + 0.984i)3-s − 0.5i·4-s + (0.444 + 1.07i)5-s + (0.580 + 0.404i)6-s + (0.567 + 0.235i)7-s + (−0.250 − 0.250i)8-s + (−0.938 + 0.346i)9-s + (0.758 + 0.314i)10-s + (−1.32 − 0.550i)11-s + (0.492 − 0.0879i)12-s + 1.50i·13-s + (0.401 − 0.166i)14-s + (−0.978 + 0.626i)15-s − 0.250·16-s + (0.952 + 0.305i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $0.208 - 0.978i$
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: \((2,\ 102,\ (\ :2),\ 0.208 - 0.978i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.80233 + 1.45893i\)
\(L(\frac12)\) \(\approx\) \(1.80233 + 1.45893i\)
\(L(3)\) 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 \( 1 + (-1.58 - 8.85i)T \)
17 \( 1 + (-275. - 88.3i)T \)
good5 \( 1 + (-11.1 - 26.8i)T + (-441. + 441. i)T^{2} \)
7 \( 1 + (-27.8 - 11.5i)T + (1.69e3 + 1.69e3i)T^{2} \)
11 \( 1 + (160. + 66.6i)T + (1.03e4 + 1.03e4i)T^{2} \)
13 \( 1 - 253. iT - 2.85e4T^{2} \)
19 \( 1 + (-230. - 230. i)T + 1.30e5iT^{2} \)
23 \( 1 + (-100. - 41.4i)T + (1.97e5 + 1.97e5i)T^{2} \)
29 \( 1 + (292. + 705. i)T + (-5.00e5 + 5.00e5i)T^{2} \)
31 \( 1 + (-204. - 493. i)T + (-6.53e5 + 6.53e5i)T^{2} \)
37 \( 1 + (-639. - 1.54e3i)T + (-1.32e6 + 1.32e6i)T^{2} \)
41 \( 1 + (-336. + 811. i)T + (-1.99e6 - 1.99e6i)T^{2} \)
43 \( 1 + (-2.23e3 + 2.23e3i)T - 3.41e6iT^{2} \)
47 \( 1 - 1.92e3T + 4.87e6T^{2} \)
53 \( 1 + (1.14e3 - 1.14e3i)T - 7.89e6iT^{2} \)
59 \( 1 + (2.92e3 + 2.92e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (5.17e3 + 2.14e3i)T + (9.79e6 + 9.79e6i)T^{2} \)
67 \( 1 - 6.24e3T + 2.01e7T^{2} \)
71 \( 1 + (337. - 139. i)T + (1.79e7 - 1.79e7i)T^{2} \)
73 \( 1 + (-8.50e3 + 3.52e3i)T + (2.00e7 - 2.00e7i)T^{2} \)
79 \( 1 + (-741. + 1.79e3i)T + (-2.75e7 - 2.75e7i)T^{2} \)
83 \( 1 + (-3.33e3 + 3.33e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 9.07e3T + 6.27e7T^{2} \)
97 \( 1 + (-1.18e3 + 489. i)T + (6.25e7 - 6.25e7i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.80879115712529817056612790078, −12.03904000451290563326255424853, −11.02572209068647629987845906577, −10.37759508334073033686090007079, −9.370774753087891811717963790802, −7.88748111309821055384643103815, −6.09850184571407445794862597244, −5.01154269196252659124473096780, −3.50475039227693886738432375427, −2.31033156670282336599294632508, 0.915921326488649258657808445270, 2.77376283251494891160319308159, 5.01839278228991818539176864684, 5.69038551602898018033414539785, 7.54415401170735717642460085669, 7.962193517070002176958159433804, 9.342170733962543196886736239933, 10.94801968711134112929042351414, 12.50526961169526374524759142877, 12.83385579807848636892553131190

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