Properties

Label 2-1216-76.75-c1-0-10
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  + 5-s + 4.35i·7-s − 3·9-s − 4.35i·11-s + 7·17-s + 4.35i·19-s + 8.71i·23-s − 4·25-s + 4.35i·35-s + 13.0i·43-s − 3·45-s + 4.35i·47-s − 12.0·49-s − 4.35i·55-s − 15·61-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.64i·7-s − 9-s − 1.31i·11-s + 1.69·17-s + 0.999i·19-s + 1.81i·23-s − 0.800·25-s + 0.736i·35-s + 1.99i·43-s − 0.447·45-s + 0.635i·47-s − 1.71·49-s − 0.587i·55-s − 1.92·61-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

\(L(1)\) \(\approx\) \(1.419278522\)
\(L(\frac12)\) \(\approx\) \(1.419278522\)
\(L(\frac{3}{2})\) 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 \( 1 - T + 5T^{2} \)
7 \( 1 - 4.35iT - 7T^{2} \)
11 \( 1 + 4.35iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
23 \( 1 - 8.71iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 13.0iT - 43T^{2} \)
47 \( 1 - 4.35iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
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

−9.598640470651873021927810079484, −9.289689289756409509468176085632, −8.182529881711780159313974762446, −7.892461696995965956192970964650, −6.05933519784711203569568899208, −5.84570053584007553397729896467, −5.27621839607647452944859918194, −3.44894845950579846685311982810, −2.88082250231297329570457088281, −1.54550165955808104194140134970, 0.61090724662325347051808232074, 2.10492350651637610444647579425, 3.33993944117661109114552992065, 4.36506464712612395828591044825, 5.19137696551245104377824290314, 6.27600795084745381564055865545, 7.14724539298870820808912222030, 7.72824453972356422499019613261, 8.730652112905267376001911698192, 9.733159759828253484517382507585

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