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Properties

Label 2-102-51.8-c4-0-14
Degree $2$
Conductor $102$
Sign $0.0955 - 0.995i$
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 + (8.99 − 0.311i)3-s + 8i·4-s + (−13.3 + 32.3i)5-s + (18.6 + 17.3i)6-s + (79.8 − 33.0i)7-s + (−16 + 16i)8-s + (80.8 − 5.59i)9-s + (−91.4 + 37.8i)10-s + (−156. + 64.8i)11-s + (2.48 + 71.9i)12-s − 3.42i·13-s + (225. + 93.5i)14-s + (−110. + 295. i)15-s − 64·16-s + (−59.1 + 282. i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.999 − 0.0345i)3-s + 0.5i·4-s + (−0.535 + 1.29i)5-s + (0.516 + 0.482i)6-s + (1.62 − 0.674i)7-s + (−0.250 + 0.250i)8-s + (0.997 − 0.0691i)9-s + (−0.914 + 0.378i)10-s + (−1.29 + 0.536i)11-s + (0.0172 + 0.499i)12-s − 0.0202i·13-s + (1.15 + 0.477i)14-s + (−0.490 + 1.31i)15-s − 0.250·16-s + (−0.204 + 0.978i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0955 - 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0955 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $0.0955 - 0.995i$
Analytic conductor: \(10.5437\)
Root analytic conductor: \(3.24711\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{102} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :2),\ 0.0955 - 0.995i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.29999 + 2.08983i\)
\(L(\frac12)\) \(\approx\) \(2.29999 + 2.08983i\)
\(L(3)\) 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 \( 1 + (-8.99 + 0.311i)T \)
17 \( 1 + (59.1 - 282. i)T \)
good5 \( 1 + (13.3 - 32.3i)T + (-441. - 441. i)T^{2} \)
7 \( 1 + (-79.8 + 33.0i)T + (1.69e3 - 1.69e3i)T^{2} \)
11 \( 1 + (156. - 64.8i)T + (1.03e4 - 1.03e4i)T^{2} \)
13 \( 1 + 3.42iT - 2.85e4T^{2} \)
19 \( 1 + (-46.0 + 46.0i)T - 1.30e5iT^{2} \)
23 \( 1 + (-551. + 228. i)T + (1.97e5 - 1.97e5i)T^{2} \)
29 \( 1 + (-238. + 575. i)T + (-5.00e5 - 5.00e5i)T^{2} \)
31 \( 1 + (494. - 1.19e3i)T + (-6.53e5 - 6.53e5i)T^{2} \)
37 \( 1 + (-843. + 2.03e3i)T + (-1.32e6 - 1.32e6i)T^{2} \)
41 \( 1 + (944. + 2.27e3i)T + (-1.99e6 + 1.99e6i)T^{2} \)
43 \( 1 + (583. + 583. i)T + 3.41e6iT^{2} \)
47 \( 1 - 819.T + 4.87e6T^{2} \)
53 \( 1 + (2.35e3 + 2.35e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-2.35e3 + 2.35e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (1.28e3 - 531. i)T + (9.79e6 - 9.79e6i)T^{2} \)
67 \( 1 + 3.53e3T + 2.01e7T^{2} \)
71 \( 1 + (-2.31e3 - 958. i)T + (1.79e7 + 1.79e7i)T^{2} \)
73 \( 1 + (-6.57e3 - 2.72e3i)T + (2.00e7 + 2.00e7i)T^{2} \)
79 \( 1 + (222. + 536. i)T + (-2.75e7 + 2.75e7i)T^{2} \)
83 \( 1 + (4.83e3 + 4.83e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.02e4T + 6.27e7T^{2} \)
97 \( 1 + (5.33e3 + 2.20e3i)T + (6.25e7 + 6.25e7i)T^{2} \)
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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

−13.71982176966713896450399853156, −12.59722773827100983222198766068, −11.03325907845952675615183197667, −10.44412134666339676444441890773, −8.476654344040165502031019167533, −7.63086314414338907934935026250, −7.02783276506894527266796604655, −4.92208845399189459374897768061, −3.70998111265758726623576695391, −2.26019507415989781017536557318, 1.27450109584301164343743351970, 2.79175867655643995017468929173, 4.61608109546511369007282103666, 5.18505404359110749730707596412, 7.76396558967747653916140107554, 8.418896146066590167373885945775, 9.394178300191085612439677937545, 11.00940602031113660930085568248, 11.90006800381287570320468295878, 12.99478680965009816684811262105

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