Properties

Label 2-1216-152.75-c1-0-32
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  − 3.04i·5-s − 5.27i·7-s + 3·9-s + 6.50·11-s − 7.27·17-s + 4.35·19-s + 4i·23-s − 4.27·25-s − 16.0·35-s + 5.67·43-s − 9.13i·45-s + 2.72i·47-s − 20.8·49-s − 19.8i·55-s + 10.8i·61-s + ⋯
L(s)  = 1  − 1.36i·5-s − 1.99i·7-s + 9-s + 1.96·11-s − 1.76·17-s + 1.00·19-s + 0.834i·23-s − 0.854·25-s − 2.71·35-s + 0.865·43-s − 1.36i·45-s + 0.397i·47-s − 2.97·49-s − 2.67i·55-s + 1.38i·61-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: $\chi_{1216} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.904690571\)
\(L(\frac12)\) \(\approx\) \(1.904690571\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.35T \)
good3 \( 1 - 3T^{2} \)
5 \( 1 + 3.04iT - 5T^{2} \)
7 \( 1 + 5.27iT - 7T^{2} \)
11 \( 1 - 6.50T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 7.27T + 17T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 5.67T + 43T^{2} \)
47 \( 1 - 2.72iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 10.8iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 5.82T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8.71T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.365405257597178219175939964424, −8.942535280941519484153511228276, −7.70388633348990962944087145668, −7.07244341123600810908454005073, −6.37353295896070616251461915065, −4.87913046191056776700861523970, −4.19340999346431843366418750335, −3.81443903747514262171292861300, −1.48741051580411768737481763335, −0.944978688094843935553310537779, 1.82195675813168942972789442970, 2.68079062776598013186740830543, 3.76330034538093440298926728770, 4.81343188185027416642311696102, 6.15209206532998145784463642928, 6.52467784972191794386145344525, 7.25452009210317472523627177231, 8.574628282386600801058380621169, 9.196801548246484707035156648990, 9.775214285472108847226613744250

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