Properties

Label 2-1216-152.109-c0-0-1
Degree $2$
Conductor $1216$
Sign $0.786 + 0.617i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.118 − 0.673i)3-s + (0.500 + 0.181i)9-s + (0.300 + 0.173i)11-s + (0.939 − 0.342i)17-s + (−0.642 − 0.766i)19-s + (0.173 + 0.984i)25-s + (0.524 − 0.907i)27-s + (0.152 − 0.181i)33-s + (−1.26 − 0.223i)41-s + (0.642 − 0.766i)43-s + (0.5 − 0.866i)49-s + (−0.118 − 0.673i)51-s + (−0.592 + 0.342i)57-s + (−1.85 + 0.673i)59-s + (1.20 + 0.439i)67-s + ⋯
L(s)  = 1  + (0.118 − 0.673i)3-s + (0.500 + 0.181i)9-s + (0.300 + 0.173i)11-s + (0.939 − 0.342i)17-s + (−0.642 − 0.766i)19-s + (0.173 + 0.984i)25-s + (0.524 − 0.907i)27-s + (0.152 − 0.181i)33-s + (−1.26 − 0.223i)41-s + (0.642 − 0.766i)43-s + (0.5 − 0.866i)49-s + (−0.118 − 0.673i)51-s + (−0.592 + 0.342i)57-s + (−1.85 + 0.673i)59-s + (1.20 + 0.439i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.786 + 0.617i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :0),\ 0.786 + 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.182560871\)
\(L(\frac12)\) \(\approx\) \(1.182560871\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (0.642 + 0.766i)T \)
good3 \( 1 + (-0.118 + 0.673i)T + (-0.939 - 0.342i)T^{2} \)
5 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.300 - 0.173i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (1.26 + 0.223i)T + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.642 + 0.766i)T + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (0.766 + 0.642i)T^{2} \)
53 \( 1 + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (1.85 - 0.673i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (-1.20 - 0.439i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (1.32 - 0.766i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.439 - 1.20i)T + (-0.766 + 0.642i)T^{2} \)
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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

−9.834016594714501912359294201761, −8.981334628341220712058825923034, −8.136948208526445834431309992274, −7.23633888599951650297752466058, −6.81453485639260019699595900529, −5.67190278255918978419477308886, −4.73655849017714614837210645442, −3.66132266914202766266300951460, −2.44055951308115548738215604526, −1.29478865325345599634721666383, 1.52592928334195601004677255390, 3.06482814512033183405177947461, 3.97372989114457644526825441913, 4.71466321575011437885619951323, 5.84616383067320958003788467591, 6.61309937859255469102360372299, 7.68111717452451771480385205943, 8.451976466288999413472719163458, 9.313566528786407954258062683547, 10.07708319696340407521860443919

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