Properties

Label 2-1216-152.53-c0-0-1
Degree $2$
Conductor $1216$
Sign $0.617 + 0.786i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :0),\ 0.617 + 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.075170819\)
\(L(\frac12)\) \(\approx\) \(1.075170819\)
\(L(1)\) 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

−9.969514042149521263956557784255, −8.939611550761479849556239271869, −8.046502811114920795933334280571, −7.30162887246990802591565473317, −6.63545079241196165008517644445, −5.68345514636668662147246012302, −4.73127912622031973335850489679, −3.62385993646434167842761526153, −2.42113612990466492892477293578, −1.13939779746055201226335974365, 1.55430523034076738506449401481, 3.13037903922762753027582431588, 3.91317771474310893001751360321, 5.07669295973534787727807711991, 5.54754355404787220953416898462, 6.83027372430058272086083227719, 7.61002286221607804535059589091, 8.423591023220750115404716779051, 9.500412228861090241070300055302, 9.985790194545231919556344350360

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