Properties

Label 2-1216-76.31-c1-0-24
Degree $2$
Conductor $1216$
Sign $0.305 + 0.952i$
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  + (−0.5 − 0.866i)3-s + (1.5 + 2.59i)5-s − 3.46i·7-s + (1 − 1.73i)9-s − 3.46i·11-s + (4.5 + 2.59i)13-s + (1.5 − 2.59i)15-s + (−1.5 − 2.59i)17-s + (−4 + 1.73i)19-s + (−2.99 + 1.73i)21-s + (4.5 + 2.59i)23-s + (−2 + 3.46i)25-s − 5·27-s + (−7.5 − 4.33i)29-s − 4·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.670 + 1.16i)5-s − 1.30i·7-s + (0.333 − 0.577i)9-s − 1.04i·11-s + (1.24 + 0.720i)13-s + (0.387 − 0.670i)15-s + (−0.363 − 0.630i)17-s + (−0.917 + 0.397i)19-s + (−0.654 + 0.377i)21-s + (0.938 + 0.541i)23-s + (−0.400 + 0.692i)25-s − 0.962·27-s + (−1.39 − 0.804i)29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.305 + 0.952i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.305 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.667472488\)
\(L(\frac12)\) \(\approx\) \(1.667472488\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - 1.73i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.5 - 2.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.5 + 4.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-7.5 + 4.33i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.5 + 6.06i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.5 + 0.866i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 + 0.866i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.5 - 7.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (-7.5 - 4.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.5 + 4.33i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.567417471854815428389699516948, −8.906913540367312830580931253511, −7.57952143238493308166712039607, −7.00858219441326338529177599117, −6.32311778038392638116919414732, −5.75115739975073853379409614386, −4.03490186099514628124440792627, −3.51936560331470584323808025152, −2.06138151206960961211917415235, −0.78678816353653160920299950479, 1.49797549079084728521024767004, 2.44365474790947137102396523328, 4.07566690105295550910967944583, 4.91461467707331657764700480005, 5.55352417425657020935876995112, 6.22964477203102781360121394165, 7.56135636124548621508590370800, 8.625195403831133538920950547536, 9.026401424049172458813536877265, 9.724165523659296014536123864329

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