Properties

Label 2-1216-76.75-c1-0-22
Degree $2$
Conductor $1216$
Sign $-0.606 + 0.794i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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  − 2.64·3-s − 1.73i·7-s + 4.00·9-s − 3.46i·11-s + 4.58i·13-s + 3·17-s + (2.64 − 3.46i)19-s + 4.58i·21-s + 5.19i·23-s − 5·25-s − 2.64·27-s + 4.58i·29-s − 5.29·31-s + 9.16i·33-s − 9.16i·37-s + ⋯
L(s)  = 1  − 1.52·3-s − 0.654i·7-s + 1.33·9-s − 1.04i·11-s + 1.27i·13-s + 0.727·17-s + (0.606 − 0.794i)19-s + 0.999i·21-s + 1.08i·23-s − 25-s − 0.509·27-s + 0.850i·29-s − 0.950·31-s + 1.59i·33-s − 1.50i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.606 + 0.794i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.606 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5075708457\)
\(L(\frac12)\) \(\approx\) \(0.5075708457\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-2.64 + 3.46i)T \)
good3 \( 1 + 2.64T + 3T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 4.58iT - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
23 \( 1 - 5.19iT - 23T^{2} \)
29 \( 1 - 4.58iT - 29T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 + 9.16iT - 37T^{2} \)
41 \( 1 + 9.16iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 4.58iT - 53T^{2} \)
59 \( 1 + 7.93T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 2.64T + 67T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 + 6.92iT - 83T^{2} \)
89 \( 1 + 18.3iT - 89T^{2} \)
97 \( 1 + 9.16iT - 97T^{2} \)
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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

−9.550585049509338354911493652625, −8.786467651307475609210866677410, −7.34591530072929514340655078190, −7.05392149383546461651205617317, −5.80847557464670492280227643174, −5.50658236669242823059194275412, −4.33786618738200936847667330287, −3.45837092799349921579896549348, −1.57696037327253331610074079133, −0.29667533820549871869466008955, 1.29127988711736300540013461096, 2.80380046174846143033175828795, 4.22782658630816803377456899954, 5.18093220608518515839480448259, 5.75601524737962496426782283631, 6.43263947452194859177311187816, 7.53670387725448463751732062462, 8.178867372038408089827836804595, 9.563672415438273879934936898177, 10.11124384934224021788887968646

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