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Properties

Label 2-102-51.38-c2-0-1
Degree $2$
Conductor $102$
Sign $0.975 + 0.218i$
Analytic cond. $2.77929$
Root an. cond. $1.66712$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + (−2.96 + 0.427i)3-s + 2.00·4-s + (−1.26 − 1.26i)5-s + (4.19 − 0.603i)6-s + (2.01 + 2.01i)7-s − 2.82·8-s + (8.63 − 2.53i)9-s + (1.78 + 1.78i)10-s + (8.23 − 8.23i)11-s + (−5.93 + 0.854i)12-s + 22.6·13-s + (−2.85 − 2.85i)14-s + (4.28 + 3.20i)15-s + 4.00·16-s + (8.81 + 14.5i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.989 + 0.142i)3-s + 0.500·4-s + (−0.252 − 0.252i)5-s + (0.699 − 0.100i)6-s + (0.288 + 0.288i)7-s − 0.353·8-s + (0.959 − 0.281i)9-s + (0.178 + 0.178i)10-s + (0.748 − 0.748i)11-s + (−0.494 + 0.0711i)12-s + 1.74·13-s + (−0.203 − 0.203i)14-s + (0.285 + 0.213i)15-s + 0.250·16-s + (0.518 + 0.855i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $0.975 + 0.218i$
Analytic conductor: \(2.77929\)
Root analytic conductor: \(1.66712\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{102} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :1),\ 0.975 + 0.218i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.771104 - 0.0852639i\)
\(L(\frac12)\) \(\approx\) \(0.771104 - 0.0852639i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (2.96 - 0.427i)T \)
17 \( 1 + (-8.81 - 14.5i)T \)
good5 \( 1 + (1.26 + 1.26i)T + 25iT^{2} \)
7 \( 1 + (-2.01 - 2.01i)T + 49iT^{2} \)
11 \( 1 + (-8.23 + 8.23i)T - 121iT^{2} \)
13 \( 1 - 22.6T + 169T^{2} \)
19 \( 1 + 19.5iT - 361T^{2} \)
23 \( 1 + (6.84 - 6.84i)T - 529iT^{2} \)
29 \( 1 + (-8.85 - 8.85i)T + 841iT^{2} \)
31 \( 1 + (16.6 - 16.6i)T - 961iT^{2} \)
37 \( 1 + (-23.1 + 23.1i)T - 1.36e3iT^{2} \)
41 \( 1 + (-17.3 + 17.3i)T - 1.68e3iT^{2} \)
43 \( 1 - 53.6iT - 1.84e3T^{2} \)
47 \( 1 + 12.1iT - 2.20e3T^{2} \)
53 \( 1 - 58.5T + 2.80e3T^{2} \)
59 \( 1 + 3.52T + 3.48e3T^{2} \)
61 \( 1 + (-21.2 - 21.2i)T + 3.72e3iT^{2} \)
67 \( 1 - 29.5T + 4.48e3T^{2} \)
71 \( 1 + (63.8 + 63.8i)T + 5.04e3iT^{2} \)
73 \( 1 + (54.3 - 54.3i)T - 5.32e3iT^{2} \)
79 \( 1 + (-31.8 - 31.8i)T + 6.24e3iT^{2} \)
83 \( 1 - 104.T + 6.88e3T^{2} \)
89 \( 1 + 74.8iT - 7.92e3T^{2} \)
97 \( 1 + (97.0 - 97.0i)T - 9.40e3iT^{2} \)
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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

−13.36103195413996490759052084556, −12.13983770488112762044557272653, −11.27626578518682821958259732624, −10.59369794573924627010306435692, −9.134760957515190441779021333993, −8.222646211190478176756252978297, −6.61714143021528212354295987540, −5.69192106894442019779240727483, −3.90465961793458383821313048266, −1.10154474775611045534505489332, 1.30566393696953867072856723394, 4.00082930421578112533522210649, 5.79920878787893805087526965562, 6.90811815439121823932328809364, 7.956311932551498605051266450071, 9.457697726148757912994486630781, 10.55610094241979122189535495219, 11.40653305884994557198284899440, 12.18313977837001261583634732185, 13.51646213580291969896665670954

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