Properties

Label 2-1216-76.75-c1-0-11
Degree $2$
Conductor $1216$
Sign $-i$
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  + 3.27·5-s + 0.418i·7-s − 3·9-s + 6.50i·11-s − 7.27·17-s + 4.35i·19-s + 8.71i·23-s + 5.72·25-s + 1.37i·35-s − 5.67i·43-s − 9.82·45-s − 13.4i·47-s + 6.82·49-s + 21.3i·55-s + 11.2·61-s + ⋯
L(s)  = 1  + 1.46·5-s + 0.158i·7-s − 9-s + 1.96i·11-s − 1.76·17-s + 0.999i·19-s + 1.81i·23-s + 1.14·25-s + 0.231i·35-s − 0.865i·43-s − 1.46·45-s − 1.96i·47-s + 0.974·49-s + 2.87i·55-s + 1.44·61-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.597704003\)
\(L(\frac12)\) \(\approx\) \(1.597704003\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.35iT \)
good3 \( 1 + 3T^{2} \)
5 \( 1 - 3.27T + 5T^{2} \)
7 \( 1 - 0.418iT - 7T^{2} \)
11 \( 1 - 6.50iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 7.27T + 17T^{2} \)
23 \( 1 - 8.71iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 5.67iT - 43T^{2} \)
47 \( 1 + 13.4iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 5.82T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8.71iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.831980300858860564174184011178, −9.275581923933141761895357381464, −8.512145742406747764837561753160, −7.30993388739700851338498588213, −6.59112609498170056735028018995, −5.64201792171364101788997702964, −5.07908684880304459973525029547, −3.84917689826698829926305016667, −2.29995058278422064440349005770, −1.88869366439457487855130310765, 0.63074735473076003572080596253, 2.35283976722074823095843499678, 2.95131376303233461609682051183, 4.46217021859284210959720279501, 5.47996785995750485803002956296, 6.23638268251933107140580362478, 6.62695190903962523614635600408, 8.207140556315168902042586008085, 8.862546313361016445414075410385, 9.259568933484379987769908533645

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