Properties

Label 2-1216-152.29-c0-0-0
Degree $2$
Conductor $1216$
Sign $0.959 - 0.281i$
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  + (−1.50 + 1.26i)3-s + (0.500 − 2.83i)9-s + (1.32 − 0.766i)11-s + (−0.173 − 0.984i)17-s + (−0.342 − 0.939i)19-s + (0.766 + 0.642i)25-s + (1.85 + 3.20i)27-s + (−1.03 + 2.83i)33-s + (0.439 + 0.524i)41-s + (0.342 − 0.939i)43-s + (0.5 + 0.866i)49-s + (1.50 + 1.26i)51-s + (1.70 + 0.984i)57-s + (−0.223 − 1.26i)59-s + (−0.118 + 0.673i)67-s + ⋯
L(s)  = 1  + (−1.50 + 1.26i)3-s + (0.500 − 2.83i)9-s + (1.32 − 0.766i)11-s + (−0.173 − 0.984i)17-s + (−0.342 − 0.939i)19-s + (0.766 + 0.642i)25-s + (1.85 + 3.20i)27-s + (−1.03 + 2.83i)33-s + (0.439 + 0.524i)41-s + (0.342 − 0.939i)43-s + (0.5 + 0.866i)49-s + (1.50 + 1.26i)51-s + (1.70 + 0.984i)57-s + (−0.223 − 1.26i)59-s + (−0.118 + 0.673i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.959 - 0.281i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :0),\ 0.959 - 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6694147326\)
\(L(\frac12)\) \(\approx\) \(0.6694147326\)
\(L(1)\) 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 \( 1 + (0.342 + 0.939i)T \)
good3 \( 1 + (1.50 - 1.26i)T + (0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.32 + 0.766i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.439 - 0.524i)T + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.342 + 0.939i)T + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.223 + 1.26i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.118 - 0.673i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-1.62 - 0.939i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.673 - 0.118i)T + (0.939 - 0.342i)T^{2} \)
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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

−10.02036443937041995306462382893, −9.215085877188752380496596817711, −8.857736404653668686460676085550, −7.10953149754410534283276758894, −6.47199588614834529112290557857, −5.68425049102192360720198570067, −4.84529431911140568271770440989, −4.12457433277935380765076252523, −3.16764822558467111678555864723, −0.874569412338624668585024776104, 1.23910926268359474326035321563, 2.12104111770290365155101052704, 4.05527060704066262215472225775, 4.92938352888714180568780801509, 6.08400663843026785017736672166, 6.38263135896529070721597018102, 7.24139452214718781095818405374, 7.979884364153337119762600971611, 8.998993558701019896141797040433, 10.27641339167269573519435032129

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