Properties

Label 2-1216-8.5-c1-0-22
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  + 3.37i·3-s − 2.52i·5-s + 3.31·7-s − 8.37·9-s − 2.37i·11-s − 5.84i·13-s + 8.51·15-s + 5·17-s i·19-s + 11.1i·21-s + 0.792·23-s − 1.37·25-s − 18.1i·27-s + 2.67i·29-s + 3.46·31-s + ⋯
L(s)  = 1  + 1.94i·3-s − 1.12i·5-s + 1.25·7-s − 2.79·9-s − 0.715i·11-s − 1.61i·13-s + 2.19·15-s + 1.21·17-s − 0.229i·19-s + 2.44i·21-s + 0.165·23-s − 0.274·25-s − 3.48i·27-s + 0.496i·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

\(L(1)\) \(\approx\) \(1.762880615\)
\(L(\frac12)\) \(\approx\) \(1.762880615\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.37iT - 3T^{2} \)
5 \( 1 + 2.52iT - 5T^{2} \)
7 \( 1 - 3.31T + 7T^{2} \)
11 \( 1 + 2.37iT - 11T^{2} \)
13 \( 1 + 5.84iT - 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
23 \( 1 - 0.792T + 23T^{2} \)
29 \( 1 - 2.67iT - 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 + 10.0iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 3.62iT - 43T^{2} \)
47 \( 1 + 0.644T + 47T^{2} \)
53 \( 1 + 6.13iT - 53T^{2} \)
59 \( 1 + 1.37iT - 59T^{2} \)
61 \( 1 - 14.5iT - 61T^{2} \)
67 \( 1 + 2.62iT - 67T^{2} \)
71 \( 1 + 6.63T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + 0.294T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + 2.74T + 97T^{2} \)
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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

−9.793453424978831950639587956594, −8.935056255208860080553747564614, −8.364372323756952524397053718469, −7.83517240208153323611220012322, −5.72853265142783018441458397021, −5.34130589979314354954984965027, −4.74988223034444358934305836803, −3.80768512251790735407297090530, −2.87994764277527998292431646928, −0.843841788757877423698514414155, 1.39862032517824890457937583829, 2.07375692236326742150565582249, 3.09818265632265382456473813380, 4.65104966035966287521323056743, 5.82186832003373871714413879937, 6.69548887093013696639102278868, 7.13710281756055274267487485458, 7.88286956828739909512687145111, 8.467105976325673024627175568916, 9.656744556135019869121598832131

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