Properties

Label 2-1216-8.5-c1-0-15
Degree $2$
Conductor $1216$
Sign $0.965 - 0.258i$
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  − 1.31i·5-s + 4.77·7-s + 3·9-s + 2.27i·11-s + 6.09i·13-s − 4.27·17-s i·19-s + 3.46·23-s + 3.27·25-s + 6.09i·29-s − 2.62·31-s − 6.27i·35-s + 0.837i·37-s − 10.5·41-s − 10.2i·43-s + ⋯
L(s)  = 1  − 0.587i·5-s + 1.80·7-s + 9-s + 0.685i·11-s + 1.68i·13-s − 1.03·17-s − 0.229i·19-s + 0.722·23-s + 0.654·25-s + 1.13i·29-s − 0.471·31-s − 1.06i·35-s + 0.137i·37-s − 1.64·41-s − 1.56i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.965 - 0.258i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.965 - 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.144418318\)
\(L(\frac12)\) \(\approx\) \(2.144418318\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 - 3T^{2} \)
5 \( 1 + 1.31iT - 5T^{2} \)
7 \( 1 - 4.77T + 7T^{2} \)
11 \( 1 - 2.27iT - 11T^{2} \)
13 \( 1 - 6.09iT - 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 6.09iT - 29T^{2} \)
31 \( 1 + 2.62T + 31T^{2} \)
37 \( 1 - 0.837iT - 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + 4.77T + 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + 8.54iT - 59T^{2} \)
61 \( 1 + 1.31iT - 61T^{2} \)
67 \( 1 + 8.54iT - 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 - 4.27T + 73T^{2} \)
79 \( 1 - 4.30T + 79T^{2} \)
83 \( 1 - 13.0iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 1.45T + 97T^{2} \)
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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

−9.583023982441507649893715781733, −8.896892215032173817562112540128, −8.290206393357214929101968119481, −7.10086906027559904031150279865, −6.83458887506654892783131065164, −4.99821949383976215696163448755, −4.82568668545160334659386918770, −3.97869620793611312304494692195, −2.00086097701687889257534285920, −1.47828214726312615245377401213, 1.09145196081869241164768145030, 2.32698484820637173328689697976, 3.52271547231018074195822402822, 4.67418640575727778964870878595, 5.29537433827877107004072744788, 6.43260325826271880264155993815, 7.36978594156801280361322015225, 8.075706565316268716914160454711, 8.626279741849461338071419563924, 9.882837459770036902588518980455

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