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Properties

Label 2-10e2-20.7-c3-0-23
Degree $2$
Conductor $100$
Sign $-0.420 - 0.907i$
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  + (0.813 − 2.70i)2-s + (−2.61 − 2.61i)3-s + (−6.67 − 4.40i)4-s + (−9.22 + 4.96i)6-s + (−17.7 + 17.7i)7-s + (−17.3 + 14.4i)8-s − 13.2i·9-s + 7.37i·11-s + (5.93 + 29.0i)12-s + (2.68 − 2.68i)13-s + (33.6 + 62.6i)14-s + (25.1 + 58.8i)16-s + (20.2 + 20.2i)17-s + (−36.0 − 10.8i)18-s − 135.·19-s + ⋯
L(s)  = 1  + (0.287 − 0.957i)2-s + (−0.503 − 0.503i)3-s + (−0.834 − 0.551i)4-s + (−0.627 + 0.337i)6-s + (−0.959 + 0.959i)7-s + (−0.767 + 0.640i)8-s − 0.492i·9-s + 0.202i·11-s + (0.142 + 0.698i)12-s + (0.0572 − 0.0572i)13-s + (0.643 + 1.19i)14-s + (0.392 + 0.919i)16-s + (0.288 + 0.288i)17-s + (−0.471 − 0.141i)18-s − 1.63·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.420 - 0.907i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ -0.420 - 0.907i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.138937 + 0.217426i\)
\(L(\frac12)\) \(\approx\) \(0.138937 + 0.217426i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.813 + 2.70i)T \)
5 \( 1 \)
good3 \( 1 + (2.61 + 2.61i)T + 27iT^{2} \)
7 \( 1 + (17.7 - 17.7i)T - 343iT^{2} \)
11 \( 1 - 7.37iT - 1.33e3T^{2} \)
13 \( 1 + (-2.68 + 2.68i)T - 2.19e3iT^{2} \)
17 \( 1 + (-20.2 - 20.2i)T + 4.91e3iT^{2} \)
19 \( 1 + 135.T + 6.85e3T^{2} \)
23 \( 1 + (71.0 + 71.0i)T + 1.21e4iT^{2} \)
29 \( 1 + 34.2iT - 2.43e4T^{2} \)
31 \( 1 + 187. iT - 2.97e4T^{2} \)
37 \( 1 + (250. + 250. i)T + 5.06e4iT^{2} \)
41 \( 1 + 211.T + 6.89e4T^{2} \)
43 \( 1 + (-46.7 - 46.7i)T + 7.95e4iT^{2} \)
47 \( 1 + (-189. + 189. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-74.5 + 74.5i)T - 1.48e5iT^{2} \)
59 \( 1 + 101.T + 2.05e5T^{2} \)
61 \( 1 - 232.T + 2.26e5T^{2} \)
67 \( 1 + (34.7 - 34.7i)T - 3.00e5iT^{2} \)
71 \( 1 - 614. iT - 3.57e5T^{2} \)
73 \( 1 + (-37.4 + 37.4i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.00e3T + 4.93e5T^{2} \)
83 \( 1 + (423. + 423. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + (-536. - 536. i)T + 9.12e5iT^{2} \)
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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

−12.48724808990495545038520893878, −11.95954062128826196373738230870, −10.67232975719616550291911854893, −9.572139048666081565856365828568, −8.563393488067257380580074718661, −6.51162144651590442469069510540, −5.67354038258085572636325455446, −3.86443166548440827820634536486, −2.23863097232206568177176720531, −0.13625570799469186725692549053, 3.61899259916999447063599974174, 4.81272520483588058043632491098, 6.13865283415391599579634206943, 7.16303293085323208023864516621, 8.467174860076825069241031355856, 9.858668262285484076228841849303, 10.68885628775523855623387088901, 12.23937901474502812419082893563, 13.35075045932997636041012501440, 14.00680668852662499291836364388

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