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Properties

Label 2-3200-8.5-c1-0-43
Degree $2$
Conductor $3200$
Sign $1$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·7-s + 3·9-s − 6.32i·11-s + 4.47i·13-s + 6.32i·19-s + 8.48·23-s + 4.47i·37-s − 2·41-s + 2.82·47-s + 1.00·49-s − 13.4i·53-s + 6.32i·59-s + 8.48·63-s − 17.8i·77-s + 9·81-s + ⋯
L(s)  = 1  + 1.06·7-s + 9-s − 1.90i·11-s + 1.24i·13-s + 1.45i·19-s + 1.76·23-s + 0.735i·37-s − 0.312·41-s + 0.412·47-s + 0.142·49-s − 1.84i·53-s + 0.823i·59-s + 1.06·63-s − 2.03i·77-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.463867752\)
\(L(\frac12)\) \(\approx\) \(2.463867752\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 - 3T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 + 6.32iT - 11T^{2} \)
13 \( 1 - 4.47iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 6.32iT - 19T^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 4.47iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 13.4iT - 53T^{2} \)
59 \( 1 - 6.32iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−8.560682535304477597938052456170, −8.070900393724353150754324746932, −7.15719296455968089707747111363, −6.46874604456386619779202677328, −5.57770383685539705263763834450, −4.80994258513261655632253113213, −3.98681574946037135357774559178, −3.17853737809823398660216616788, −1.81343494848902307280838838410, −1.06386010922047019019308053790, 1.00004350701790235168483505505, 1.98061311870984812950746836486, 2.92889693527705747627755365726, 4.26084194577213046080715369399, 4.80964536829037161654070140210, 5.30999175974862547841902186189, 6.66004029974560961851801775079, 7.37768738226385904089236422511, 7.60922898720431629227557508610, 8.722021574605619950004942109885

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