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Properties

Label 2-1617-231.164-c0-0-1
Degree $2$
Conductor $1617$
Sign $0.883 - 0.469i$
Analytic cond. $0.806988$
Root an. cond. $0.898325$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.991 − 0.130i)3-s + (0.5 − 0.866i)4-s + (0.923 + 1.60i)5-s + (0.965 + 0.258i)9-s + (0.866 + 0.5i)11-s + (−0.608 + 0.793i)12-s + (−0.707 − 1.70i)15-s + (−0.499 − 0.866i)16-s + 1.84·20-s + (−1.22 + 0.707i)23-s + (−1.20 + 2.09i)25-s + (−0.923 − 0.382i)27-s + (0.662 + 0.382i)31-s + (−0.793 − 0.608i)33-s + (0.707 − 0.707i)36-s + ⋯
L(s)  = 1  + (−0.991 − 0.130i)3-s + (0.5 − 0.866i)4-s + (0.923 + 1.60i)5-s + (0.965 + 0.258i)9-s + (0.866 + 0.5i)11-s + (−0.608 + 0.793i)12-s + (−0.707 − 1.70i)15-s + (−0.499 − 0.866i)16-s + 1.84·20-s + (−1.22 + 0.707i)23-s + (−1.20 + 2.09i)25-s + (−0.923 − 0.382i)27-s + (0.662 + 0.382i)31-s + (−0.793 − 0.608i)33-s + (0.707 − 0.707i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.883 - 0.469i$
Analytic conductor: \(0.806988\)
Root analytic conductor: \(0.898325\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (1550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :0),\ 0.883 - 0.469i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.116817731\)
\(L(\frac12)\) \(\approx\) \(1.116817731\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.991 + 0.130i)T \)
7 \( 1 \)
11 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.84iT - T^{2} \)
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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

−10.02629701452341613504866615000, −9.312478177741219887690282313951, −7.61723376611894825737772133842, −6.90187326210283073515240801863, −6.37301222532083050838974929116, −5.89342540542471440643563131775, −5.00144699639387514000346068322, −3.69498498184134795123698263746, −2.32217592220509177123361266618, −1.56714299930900655888713617329, 1.10985653740586855270080944855, 2.23717580626980785789933569050, 3.92232782509004114274730709154, 4.50410093330446978220159805983, 5.57394481993827939220053034764, 6.15457099535343120140788751372, 6.91860539765208519350620339948, 8.140895582783144203910655570507, 8.675492454278906826119583730830, 9.559806358249651999331203978916

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