Properties

Label 2-1216-76.75-c1-0-33
Degree $2$
Conductor $1216$
Sign $-0.704 + 0.709i$
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  + 1.19·3-s − 1.56·5-s + 0.868i·7-s − 1.56·9-s − 3.09i·11-s − 4.74i·13-s − 1.87·15-s − 17-s + (−3.07 + 3.09i)19-s + 1.04i·21-s − 3.96i·23-s − 2.56·25-s − 5.47·27-s + 8.45i·29-s − 4.27·31-s + ⋯
L(s)  = 1  + 0.692·3-s − 0.698·5-s + 0.328i·7-s − 0.520·9-s − 0.932i·11-s − 1.31i·13-s − 0.483·15-s − 0.242·17-s + (−0.704 + 0.709i)19-s + 0.227i·21-s − 0.825i·23-s − 0.512·25-s − 1.05·27-s + 1.57i·29-s − 0.767·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.704 + 0.709i$
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: \((2,\ 1216,\ (\ :1/2),\ -0.704 + 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7449923057\)
\(L(\frac12)\) \(\approx\) \(0.7449923057\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.07 - 3.09i)T \)
good3 \( 1 - 1.19T + 3T^{2} \)
5 \( 1 + 1.56T + 5T^{2} \)
7 \( 1 - 0.868iT - 7T^{2} \)
11 \( 1 + 3.09iT - 11T^{2} \)
13 \( 1 + 4.74iT - 13T^{2} \)
17 \( 1 + T + 17T^{2} \)
23 \( 1 + 3.96iT - 23T^{2} \)
29 \( 1 - 8.45iT - 29T^{2} \)
31 \( 1 + 4.27T + 31T^{2} \)
37 \( 1 + 3.70iT - 37T^{2} \)
41 \( 1 + 3.70iT - 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + 9.27iT - 47T^{2} \)
53 \( 1 + 1.04iT - 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 - 0.684T + 61T^{2} \)
67 \( 1 - 9.74T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 8.12T + 73T^{2} \)
79 \( 1 + 8.01T + 79T^{2} \)
83 \( 1 + 9.65iT - 83T^{2} \)
89 \( 1 + 5.79iT - 89T^{2} \)
97 \( 1 - 16.9iT - 97T^{2} \)
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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

−9.061647549898115873929987575577, −8.580288456391617876397782175322, −8.025979343351487476760388348632, −7.15437239091296632765566978379, −5.92335821168823465922958170914, −5.31287674166377855138736226415, −3.83877029815780430779955792941, −3.27996004811677026887283587861, −2.19666525930915372634763662125, −0.27093893868677562968284057942, 1.82170974936525142280637151218, 2.86408456586488144597702156272, 4.09335871624689699623579309510, 4.51784044909819092541119157363, 5.96806153308317868381592019565, 6.91767070574858640592721723215, 7.69091986526395459399244027326, 8.312380967080234817048354142500, 9.364724938224173507467157357440, 9.645171935370689604975524385699

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