Properties

Label 2-10e2-25.9-c3-0-4
Degree $2$
Conductor $100$
Sign $0.875 + 0.482i$
Analytic cond. $5.90019$
Root an. cond. $2.42903$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.55 − 1.15i)3-s + (10.9 + 2.10i)5-s − 30.1i·7-s + (−10.5 + 7.66i)9-s + (51.5 + 37.4i)11-s + (−23.8 − 32.8i)13-s + (41.4 − 5.21i)15-s + (42.1 + 13.6i)17-s + (35.4 − 109. i)19-s + (−34.7 − 107. i)21-s + (−14.3 + 19.7i)23-s + (116. + 46.1i)25-s + (−87.9 + 121. i)27-s + (2.06 + 6.36i)29-s + (−97.5 + 300. i)31-s + ⋯
L(s)  = 1  + (0.683 − 0.222i)3-s + (0.982 + 0.187i)5-s − 1.62i·7-s + (−0.390 + 0.283i)9-s + (1.41 + 1.02i)11-s + (−0.509 − 0.700i)13-s + (0.713 − 0.0897i)15-s + (0.601 + 0.195i)17-s + (0.427 − 1.31i)19-s + (−0.361 − 1.11i)21-s + (−0.130 + 0.179i)23-s + (0.929 + 0.369i)25-s + (−0.626 + 0.862i)27-s + (0.0132 + 0.0407i)29-s + (−0.565 + 1.74i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.875 + 0.482i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ 0.875 + 0.482i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.13926 - 0.550501i\)
\(L(\frac12)\) \(\approx\) \(2.13926 - 0.550501i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-10.9 - 2.10i)T \)
good3 \( 1 + (-3.55 + 1.15i)T + (21.8 - 15.8i)T^{2} \)
7 \( 1 + 30.1iT - 343T^{2} \)
11 \( 1 + (-51.5 - 37.4i)T + (411. + 1.26e3i)T^{2} \)
13 \( 1 + (23.8 + 32.8i)T + (-678. + 2.08e3i)T^{2} \)
17 \( 1 + (-42.1 - 13.6i)T + (3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (-35.4 + 109. i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + (14.3 - 19.7i)T + (-3.75e3 - 1.15e4i)T^{2} \)
29 \( 1 + (-2.06 - 6.36i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (97.5 - 300. i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (201. + 277. i)T + (-1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (356. - 259. i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 - 291. iT - 7.95e4T^{2} \)
47 \( 1 + (-100. + 32.7i)T + (8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (346. - 112. i)T + (1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (269. - 195. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (-10.8 - 7.90i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + (-54.7 - 17.7i)T + (2.43e5 + 1.76e5i)T^{2} \)
71 \( 1 + (72.3 + 222. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (-262. + 360. i)T + (-1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (17.4 + 53.8i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-1.29e3 - 421. i)T + (4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + (426. + 309. i)T + (2.17e5 + 6.70e5i)T^{2} \)
97 \( 1 + (1.27e3 - 413. i)T + (7.38e5 - 5.36e5i)T^{2} \)
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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

−13.60163782531980062129719277167, −12.51595497083045118431602405551, −10.95550577920958395880949100729, −9.992186218375082968605420367088, −9.076415832025153249057685146813, −7.52093376656711271451872388304, −6.75682663926353111886932562336, −4.94599862948718857037730467266, −3.28053967058540001022613797614, −1.50643644213445597494405324444, 1.98523379101894069062422548760, 3.44350660851471538219888826075, 5.52023874121411238708818353454, 6.30960608388160671495266886229, 8.405256287843285957105274613883, 9.119201276689612186221168194915, 9.778988742186268414145392888596, 11.67771054441583850648927591629, 12.22769553329071193900736940162, 13.85741008976797911009910985775

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