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Properties

Label 2-10e2-4.3-c6-0-25
Degree $2$
Conductor $100$
Sign $-0.103 + 0.994i$
Analytic cond. $23.0054$
Root an. cond. $4.79639$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.35 − 5.94i)2-s − 40.3i·3-s + (−6.65 + 63.6i)4-s + (−239. + 216. i)6-s + 450. i·7-s + (413. − 301. i)8-s − 900.·9-s − 390. i·11-s + (2.56e3 + 268. i)12-s + 3.23e3·13-s + (2.67e3 − 2.41e3i)14-s + (−4.00e3 − 846. i)16-s + 4.93e3·17-s + (4.82e3 + 5.35e3i)18-s + 1.98e3i·19-s + ⋯
L(s)  = 1  + (−0.669 − 0.742i)2-s − 1.49i·3-s + (−0.103 + 0.994i)4-s + (−1.11 + 1.00i)6-s + 1.31i·7-s + (0.808 − 0.588i)8-s − 1.23·9-s − 0.293i·11-s + (1.48 + 0.155i)12-s + 1.47·13-s + (0.976 − 0.879i)14-s + (−0.978 − 0.206i)16-s + 1.00·17-s + (0.827 + 0.918i)18-s + 0.288i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.103 + 0.994i$
Analytic conductor: \(23.0054\)
Root analytic conductor: \(4.79639\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3),\ -0.103 + 0.994i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.948168 - 1.05241i\)
\(L(\frac12)\) \(\approx\) \(0.948168 - 1.05241i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.35 + 5.94i)T \)
5 \( 1 \)
good3 \( 1 + 40.3iT - 729T^{2} \)
7 \( 1 - 450. iT - 1.17e5T^{2} \)
11 \( 1 + 390. iT - 1.77e6T^{2} \)
13 \( 1 - 3.23e3T + 4.82e6T^{2} \)
17 \( 1 - 4.93e3T + 2.41e7T^{2} \)
19 \( 1 - 1.98e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.07e3iT - 1.48e8T^{2} \)
29 \( 1 - 2.11e4T + 5.94e8T^{2} \)
31 \( 1 - 3.32e4iT - 8.87e8T^{2} \)
37 \( 1 + 1.41e4T + 2.56e9T^{2} \)
41 \( 1 + 4.28e4T + 4.75e9T^{2} \)
43 \( 1 + 5.77e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.38e5iT - 1.07e10T^{2} \)
53 \( 1 - 2.77e5T + 2.21e10T^{2} \)
59 \( 1 - 2.81e4iT - 4.21e10T^{2} \)
61 \( 1 + 3.40e4T + 5.15e10T^{2} \)
67 \( 1 - 2.30e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.14e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.11e4T + 1.51e11T^{2} \)
79 \( 1 - 3.65e5iT - 2.43e11T^{2} \)
83 \( 1 + 5.02e4iT - 3.26e11T^{2} \)
89 \( 1 - 1.77e5T + 4.96e11T^{2} \)
97 \( 1 - 1.15e6T + 8.32e11T^{2} \)
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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

−12.17154071749904332635050072680, −11.74828530435880590160098231533, −10.37446015485346493571854609317, −8.736821234733480956678882596077, −8.310820803094221071928958334994, −6.94519215330077594786376755280, −5.73453380645850494322219510899, −3.27558545617689061760183540256, −1.97668504348823879377656960445, −0.902693942768918303000199691805, 0.942746391031468824158838242629, 3.70596395340336784902598165035, 4.74597100392398043509317997167, 6.11515724070533409845299816149, 7.53890760042862602602313989259, 8.726085602179436894364826686745, 9.845657506393475036545678338853, 10.45246197233160808946730690839, 11.28366343364002800935882822878, 13.44184009101509849866295962039

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