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Properties

Label 2-804-12.11-c1-0-25
Degree $2$
Conductor $804$
Sign $0.997 - 0.0728i$
Analytic cond. $6.41997$
Root an. cond. $2.53376$
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  + (−1.36 − 0.366i)2-s + (−0.973 − 1.43i)3-s + (1.73 + 1.00i)4-s + 2.42i·5-s + (0.805 + 2.31i)6-s − 1.75i·7-s + (−1.99 − 2.00i)8-s + (−1.10 + 2.78i)9-s + (0.887 − 3.30i)10-s + 0.337·11-s + (−0.252 − 3.45i)12-s − 1.33·13-s + (−0.642 + 2.39i)14-s + (3.46 − 2.35i)15-s + (1.99 + 3.46i)16-s − 4.05i·17-s + ⋯
L(s)  = 1  + (−0.965 − 0.259i)2-s + (−0.562 − 0.826i)3-s + (0.865 + 0.500i)4-s + 1.08i·5-s + (0.328 + 0.944i)6-s − 0.663i·7-s + (−0.706 − 0.707i)8-s + (−0.367 + 0.929i)9-s + (0.280 − 1.04i)10-s + 0.101·11-s + (−0.0728 − 0.997i)12-s − 0.371·13-s + (−0.171 + 0.640i)14-s + (0.895 − 0.608i)15-s + (0.499 + 0.866i)16-s − 0.984i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.997 - 0.0728i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.997 - 0.0728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723920 + 0.0264142i\)
\(L(\frac12)\) \(\approx\) \(0.723920 + 0.0264142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 + (0.973 + 1.43i)T \)
67 \( 1 + iT \)
good5 \( 1 - 2.42iT - 5T^{2} \)
7 \( 1 + 1.75iT - 7T^{2} \)
11 \( 1 - 0.337T + 11T^{2} \)
13 \( 1 + 1.33T + 13T^{2} \)
17 \( 1 + 4.05iT - 17T^{2} \)
19 \( 1 - 1.16iT - 19T^{2} \)
23 \( 1 - 1.81T + 23T^{2} \)
29 \( 1 - 8.51iT - 29T^{2} \)
31 \( 1 - 4.75iT - 31T^{2} \)
37 \( 1 - 6.50T + 37T^{2} \)
41 \( 1 + 0.0765iT - 41T^{2} \)
43 \( 1 - 9.26iT - 43T^{2} \)
47 \( 1 + 8.82T + 47T^{2} \)
53 \( 1 + 2.03iT - 53T^{2} \)
59 \( 1 - 14.7T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
71 \( 1 - 3.10T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + 4.35iT - 79T^{2} \)
83 \( 1 - 8.52T + 83T^{2} \)
89 \( 1 + 0.230iT - 89T^{2} \)
97 \( 1 - 0.733T + 97T^{2} \)
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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

−10.40070541789337071100926038172, −9.614857386397920653666933548319, −8.435971156850684509691254082259, −7.50483288996815476771731917367, −6.96066684465345417250687394076, −6.44348333089389837392816235710, −5.07141538521950562956323863080, −3.36408527899435410239892499166, −2.41820649858650301697124924345, −1.01139644967234757324506499105, 0.67967726555563066066828635341, 2.33443994816604786304785299685, 4.00511059909986286772115029101, 5.14437011106339315264589365559, 5.78864771511135300414737450696, 6.69202551877501136632696279505, 8.062402076863258318148671327884, 8.668404959650568192183037447415, 9.453260994064837015859173558600, 9.956241639203442989360961196857

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