Properties

Label 2-1216-152.13-c0-0-1
Degree $2$
Conductor $1216$
Sign $0.966 - 0.258i$
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  + (1.20 + 0.439i)3-s + (0.500 + 0.419i)9-s + (1.62 − 0.939i)11-s + (−0.766 + 0.642i)17-s + (−0.984 − 0.173i)19-s + (−0.939 + 0.342i)25-s + (−0.223 − 0.386i)27-s + (2.37 − 0.419i)33-s + (−0.673 + 1.85i)41-s + (0.984 − 0.173i)43-s + (0.5 + 0.866i)49-s + (−1.20 + 0.439i)51-s + (−1.11 − 0.642i)57-s + (0.524 − 0.439i)59-s + (−1.50 − 1.26i)67-s + ⋯
L(s)  = 1  + (1.20 + 0.439i)3-s + (0.500 + 0.419i)9-s + (1.62 − 0.939i)11-s + (−0.766 + 0.642i)17-s + (−0.984 − 0.173i)19-s + (−0.939 + 0.342i)25-s + (−0.223 − 0.386i)27-s + (2.37 − 0.419i)33-s + (−0.673 + 1.85i)41-s + (0.984 − 0.173i)43-s + (0.5 + 0.866i)49-s + (−1.20 + 0.439i)51-s + (−1.11 − 0.642i)57-s + (0.524 − 0.439i)59-s + (−1.50 − 1.26i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.966 - 0.258i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :0),\ 0.966 - 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.578772951\)
\(L(\frac12)\) \(\approx\) \(1.578772951\)
\(L(1)\) 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 \( 1 + (0.984 + 0.173i)T \)
good3 \( 1 + (-1.20 - 0.439i)T + (0.766 + 0.642i)T^{2} \)
5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.673 - 1.85i)T + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (-0.984 + 0.173i)T + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.524 + 0.439i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (1.50 + 1.26i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.300 - 0.173i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \)
97 \( 1 + (1.26 + 1.50i)T + (-0.173 + 0.984i)T^{2} \)
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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

−9.664407487662097205072237535755, −9.043402984284204121653523960395, −8.566943861276778336103046211499, −7.78475671458919818679733624034, −6.55669722805347717045226531704, −5.98269373651606472665314733886, −4.39142970610450376511638233786, −3.84997157259933673593529863929, −2.92200061633886908781910854403, −1.69335723270342150790973741657, 1.71730106099862639184364120584, 2.47497633780502933382292621077, 3.79548342097692875841267563506, 4.40205678535720652037113666160, 5.85180875607562226604481835214, 6.93418245425843655548396471840, 7.31506519699841310325507612678, 8.484306293323924849265632649148, 8.938164518796250414156038944089, 9.624970354294502192386385289960

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