Properties

Label 2-1216-8.5-c1-0-19
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  + 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

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

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

\(L(1)\) \(\approx\) \(1.239869147\)
\(L(\frac12)\) \(\approx\) \(1.239869147\)
\(L(\frac{3}{2})\) 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

−9.919132998984469878675560965047, −9.013215182398201940663919415876, −8.342700137250518864841514740797, −7.29433107283647540097617675054, −5.87835628973417253256046822091, −5.64847233955444988663265826413, −4.40128284536815089414849652370, −3.49248163063783348310902555142, −2.93504849566132738532662159349, −0.59846594733550546656956410950, 1.21974153605084793001497723306, 2.37916494402469497161880132543, 3.34262500070991929730988144828, 4.61670428141634720614209008362, 5.94238110829453136697497968880, 6.71415593473401545906260977149, 7.06066985850514111188601620222, 7.87507177065541613988392613575, 8.962649301562549292287938329039, 9.767210035943485499961485880048

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