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Properties

Label 2-1216-19.7-c1-0-0
Degree $2$
Conductor $1216$
Sign $0.761 - 0.648i$
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  + (−1.32 − 2.29i)3-s + (−0.822 − 1.42i)5-s − 3.64·7-s + (−2 + 3.46i)9-s − 4.64·11-s + (1 − 1.73i)13-s + (−2.17 + 3.77i)15-s + (1.67 + 4.02i)19-s + (4.82 + 8.35i)21-s + (−0.822 + 1.42i)23-s + (1.14 − 1.98i)25-s + 2.64·27-s + (0.822 − 1.42i)29-s + 5.64·31-s + (6.14 + 10.6i)33-s + ⋯
L(s)  = 1  + (−0.763 − 1.32i)3-s + (−0.368 − 0.637i)5-s − 1.37·7-s + (−0.666 + 1.15i)9-s − 1.40·11-s + (0.277 − 0.480i)13-s + (−0.562 + 0.973i)15-s + (0.384 + 0.923i)19-s + (1.05 + 1.82i)21-s + (−0.171 + 0.297i)23-s + (0.229 − 0.396i)25-s + 0.509·27-s + (0.152 − 0.264i)29-s + 1.01·31-s + (1.06 + 1.85i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.761 - 0.648i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1855209766\)
\(L(\frac12)\) \(\approx\) \(0.1855209766\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 - 4.02i)T \)
good3 \( 1 + (1.32 + 2.29i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.822 + 1.42i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.64T + 7T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.822 - 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.822 + 1.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.64T + 31T^{2} \)
37 \( 1 + 0.354T + 37T^{2} \)
41 \( 1 + (-0.145 - 0.252i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.64 + 9.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.17 + 3.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.29 - 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.96 - 6.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.468 + 0.811i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.322 - 0.559i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.35 + 2.34i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.854 + 1.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.93T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.85 - 3.21i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.05736939915223475077844252822, −8.818000534258343748284367562580, −7.946103292611499350758293663691, −7.40892613305471491506957425827, −6.40090979781779350893909311950, −5.83629998506531466732159568443, −4.98751365359077016470747333256, −3.54938883866320437045529466784, −2.43572757931179661472900819619, −0.913235410893801392131927457421, 0.11277969516880918091749344581, 2.82029558747816341376073781711, 3.42579358950809774759942899864, 4.53658887281683424840006213977, 5.28039295411706622458694157527, 6.28569941038167984748345209221, 6.90800148141519126680685257146, 8.039027582216312493823047667267, 9.170056317002759494094575939981, 9.874645624636152980063881012221

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