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Properties

Label 2-48e2-12.11-c1-0-21
Degree $2$
Conductor $2304$
Sign $0.577 + 0.816i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
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  − 1.41i·5-s + 6·13-s − 4.24i·17-s + 2.99·25-s + 9.89i·29-s − 12·37-s − 12.7i·41-s + 7·49-s − 7.07i·53-s + 12·61-s − 8.48i·65-s + 16·73-s − 6·85-s − 4.24i·89-s + 8·97-s + ⋯
L(s)  = 1  − 0.632i·5-s + 1.66·13-s − 1.02i·17-s + 0.599·25-s + 1.83i·29-s − 1.97·37-s − 1.98i·41-s + 49-s − 0.971i·53-s + 1.53·61-s − 1.05i·65-s + 1.87·73-s − 0.650·85-s − 0.449i·89-s + 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (2303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.866192474\)
\(L(\frac12)\) \(\approx\) \(1.866192474\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 1.41iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 12T + 37T^{2} \)
41 \( 1 + 12.7iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 16T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 4.24iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−8.665448327469721039064885177842, −8.520252027462985877460482464529, −7.18987703843219766734943236132, −6.71844592878880321149052682834, −5.50993757594403209479604473140, −5.10944623049805629349569752524, −3.92845347855090775718477750503, −3.22571441015524043697539923249, −1.83646560878485289233767997593, −0.75773277705540921345974271839, 1.16226903652818872137226635763, 2.36249069848300583001755989159, 3.48144293144950964823622845660, 4.06305588673099705790410299508, 5.25931563034502464343249746713, 6.26246697919264591378252245852, 6.53037306920614621635227014891, 7.68150653352053951624202895369, 8.358302792435375332841741510610, 8.991704092772574245798637408766

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