Properties

Label 2-1617-77.76-c1-0-61
Degree $2$
Conductor $1617$
Sign $-0.535 + 0.844i$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
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  − 1.07i·2-s + i·3-s + 0.842·4-s − 3.36i·5-s + 1.07·6-s − 3.05i·8-s − 9-s − 3.61·10-s + (−3.27 + 0.491i)11-s + 0.842i·12-s + 4.53·13-s + 3.36·15-s − 1.60·16-s + 1.76·17-s + 1.07i·18-s + 7.04·19-s + ⋯
L(s)  = 1  − 0.760i·2-s + 0.577i·3-s + 0.421·4-s − 1.50i·5-s + 0.439·6-s − 1.08i·8-s − 0.333·9-s − 1.14·10-s + (−0.988 + 0.148i)11-s + 0.243i·12-s + 1.25·13-s + 0.868·15-s − 0.401·16-s + 0.427·17-s + 0.253i·18-s + 1.61·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.535 + 0.844i$
Analytic conductor: \(12.9118\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ -0.535 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.965202127\)
\(L(\frac12)\) \(\approx\) \(1.965202127\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 \)
11 \( 1 + (3.27 - 0.491i)T \)
good2 \( 1 + 1.07iT - 2T^{2} \)
5 \( 1 + 3.36iT - 5T^{2} \)
13 \( 1 - 4.53T + 13T^{2} \)
17 \( 1 - 1.76T + 17T^{2} \)
19 \( 1 - 7.04T + 19T^{2} \)
23 \( 1 - 3.35T + 23T^{2} \)
29 \( 1 + 3.85iT - 29T^{2} \)
31 \( 1 - 3.22iT - 31T^{2} \)
37 \( 1 + 9.04T + 37T^{2} \)
41 \( 1 + 6.67T + 41T^{2} \)
43 \( 1 + 10.7iT - 43T^{2} \)
47 \( 1 + 5.21iT - 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 + 4.52iT - 59T^{2} \)
61 \( 1 - 3.44T + 61T^{2} \)
67 \( 1 - 0.302T + 67T^{2} \)
71 \( 1 - 2.04T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 0.790iT - 79T^{2} \)
83 \( 1 + 1.19T + 83T^{2} \)
89 \( 1 - 8.59iT - 89T^{2} \)
97 \( 1 - 5.01iT - 97T^{2} \)
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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

−9.253099849977403409817826578580, −8.525027580450674893511197140470, −7.76098156750325108405709221102, −6.68276081742950169582453221059, −5.39933437028256356111251842906, −5.11775483426561728502998337946, −3.82319519151935323682419352406, −3.18278742214010895278607778549, −1.78620741204985925465236726756, −0.77498912254526934294813415692, 1.57595807043327386000416423365, 2.94771486306800844972958873742, 3.24475616219444158710814868971, 5.15472353911343830134068850590, 5.87057982905240610243289043175, 6.59210810589697388177013943527, 7.20465606110785004932070789486, 7.79172003762601088533672318151, 8.469880691018743088984007081312, 9.692448774674302138226921997050

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