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Properties

Label 2-102-51.38-c2-0-3
Degree $2$
Conductor $102$
Sign $0.887 - 0.461i$
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 + (−2.99 + 0.0213i)3-s + 2.00·4-s + (2.03 + 2.03i)5-s + (−4.24 + 0.0301i)6-s + (8.34 + 8.34i)7-s + 2.82·8-s + (8.99 − 0.128i)9-s + (2.88 + 2.88i)10-s + (10.9 − 10.9i)11-s + (−5.99 + 0.0426i)12-s − 15.6·13-s + (11.8 + 11.8i)14-s + (−6.16 − 6.07i)15-s + 4.00·16-s + (−16.6 − 3.21i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.999 + 0.00711i)3-s + 0.500·4-s + (0.407 + 0.407i)5-s + (−0.707 + 0.00502i)6-s + (1.19 + 1.19i)7-s + 0.353·8-s + (0.999 − 0.0142i)9-s + (0.288 + 0.288i)10-s + (0.996 − 0.996i)11-s + (−0.499 + 0.00355i)12-s − 1.20·13-s + (0.843 + 0.843i)14-s + (−0.410 − 0.405i)15-s + 0.250·16-s + (−0.981 − 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $0.887 - 0.461i$
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: \((2,\ 102,\ (\ :1),\ 0.887 - 0.461i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.67585 + 0.409954i\)
\(L(\frac12)\) \(\approx\) \(1.67585 + 0.409954i\)
\(L(2)\) 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 \( 1 + (2.99 - 0.0213i)T \)
17 \( 1 + (16.6 + 3.21i)T \)
good5 \( 1 + (-2.03 - 2.03i)T + 25iT^{2} \)
7 \( 1 + (-8.34 - 8.34i)T + 49iT^{2} \)
11 \( 1 + (-10.9 + 10.9i)T - 121iT^{2} \)
13 \( 1 + 15.6T + 169T^{2} \)
19 \( 1 - 26.5iT - 361T^{2} \)
23 \( 1 + (3.39 - 3.39i)T - 529iT^{2} \)
29 \( 1 + (22.7 + 22.7i)T + 841iT^{2} \)
31 \( 1 + (-34.2 + 34.2i)T - 961iT^{2} \)
37 \( 1 + (-9.99 + 9.99i)T - 1.36e3iT^{2} \)
41 \( 1 + (-11.9 + 11.9i)T - 1.68e3iT^{2} \)
43 \( 1 + 14.8iT - 1.84e3T^{2} \)
47 \( 1 + 16.3iT - 2.20e3T^{2} \)
53 \( 1 + 53.4T + 2.80e3T^{2} \)
59 \( 1 + 76.8T + 3.48e3T^{2} \)
61 \( 1 + (2.85 + 2.85i)T + 3.72e3iT^{2} \)
67 \( 1 + 55.2T + 4.48e3T^{2} \)
71 \( 1 + (-22.0 - 22.0i)T + 5.04e3iT^{2} \)
73 \( 1 + (-31.2 + 31.2i)T - 5.32e3iT^{2} \)
79 \( 1 + (-49.2 - 49.2i)T + 6.24e3iT^{2} \)
83 \( 1 - 23.5T + 6.88e3T^{2} \)
89 \( 1 - 90.4iT - 7.92e3T^{2} \)
97 \( 1 + (67.7 - 67.7i)T - 9.40e3iT^{2} \)
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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

−13.77302557835303593768431159289, −12.26933224310191241128127201164, −11.73965262133704946645089502483, −10.91381166383840613176331260289, −9.538734271718499273772685956235, −7.939782501321919973911780948482, −6.35055910178490211860342251690, −5.63056357786979906179159487008, −4.38102720579164811383706315810, −2.11096655025811537417902417436, 1.55040301083751285286175172968, 4.54520813545582280822327065760, 4.84068483945020796864899372425, 6.67089442630579266324985641842, 7.42553180507999612379105860704, 9.424250007170794051587803254285, 10.67026976827858819585481306231, 11.46529205472389258405021224066, 12.45573837346574305287206602130, 13.41314413252162383464554574742

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