Properties

Label 2-1216-8.5-c1-0-12
Degree $2$
Conductor $1216$
Sign $-0.258 - 0.965i$
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  + 2.37i·3-s + 0.792i·5-s + 3.31·7-s − 2.62·9-s − 3.37i·11-s + 4.10i·13-s − 1.87·15-s + 5·17-s + i·19-s + 7.86i·21-s + 2.52·23-s + 4.37·25-s + 0.883i·27-s + 5.98i·29-s − 3.46·31-s + ⋯
L(s)  = 1  + 1.36i·3-s + 0.354i·5-s + 1.25·7-s − 0.875·9-s − 1.01i·11-s + 1.13i·13-s − 0.485·15-s + 1.21·17-s + 0.229i·19-s + 1.71i·21-s + 0.526·23-s + 0.874·25-s + 0.169i·27-s + 1.11i·29-s − 0.622·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.258 - 0.965i$
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: \((2,\ 1216,\ (\ :1/2),\ -0.258 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.953614297\)
\(L(\frac12)\) \(\approx\) \(1.953614297\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.37iT - 3T^{2} \)
5 \( 1 - 0.792iT - 5T^{2} \)
7 \( 1 - 3.31T + 7T^{2} \)
11 \( 1 + 3.37iT - 11T^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
23 \( 1 - 2.52T + 23T^{2} \)
29 \( 1 - 5.98iT - 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 - 3.16iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 9.37iT - 43T^{2} \)
47 \( 1 + 9.30T + 47T^{2} \)
53 \( 1 + 9.45iT - 53T^{2} \)
59 \( 1 + 4.37iT - 59T^{2} \)
61 \( 1 - 4.55iT - 61T^{2} \)
67 \( 1 - 8.37iT - 67T^{2} \)
71 \( 1 + 6.63T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 - 7.48T + 89T^{2} \)
97 \( 1 - 8.74T + 97T^{2} \)
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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

−10.14314906167616906118186308215, −9.058625037482359487585022235353, −8.640604967150968629236848381858, −7.62522358784906160477305884846, −6.62109541026131474306171422011, −5.36460901845087189259938218961, −4.93254056768507418073777074947, −3.86819229434019105338563217548, −3.13392879742463735289517653005, −1.52143263540541849284876611229, 0.962322197851378193396681577292, 1.78878215865209644962971452947, 2.94361977816303228611969455357, 4.52649546749383743508688380858, 5.27592433553822880920106061045, 6.21080104298761349758029663455, 7.33395295329179552912041371936, 7.75130529194113109539068316317, 8.311172809260085018376733797215, 9.404979390538569676937055665047

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