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Properties

Label 2-1617-231.164-c0-0-8
Degree $2$
Conductor $1617$
Sign $0.448 + 0.893i$
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  + (0.793 − 0.608i)3-s + (0.5 − 0.866i)4-s + (0.382 + 0.662i)5-s + (0.258 − 0.965i)9-s + (−0.866 − 0.5i)11-s + (−0.130 − 0.991i)12-s + (0.707 + 0.292i)15-s + (−0.499 − 0.866i)16-s + 0.765·20-s + (−1.22 + 0.707i)23-s + (0.207 − 0.358i)25-s + (−0.382 − 0.923i)27-s + (1.60 + 0.923i)31-s + (−0.991 + 0.130i)33-s + (−0.707 − 0.707i)36-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)3-s + (0.5 − 0.866i)4-s + (0.382 + 0.662i)5-s + (0.258 − 0.965i)9-s + (−0.866 − 0.5i)11-s + (−0.130 − 0.991i)12-s + (0.707 + 0.292i)15-s + (−0.499 − 0.866i)16-s + 0.765·20-s + (−1.22 + 0.707i)23-s + (0.207 − 0.358i)25-s + (−0.382 − 0.923i)27-s + (1.60 + 0.923i)31-s + (−0.991 + 0.130i)33-s + (−0.707 − 0.707i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.448 + 0.893i$
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: \((2,\ 1617,\ (\ :0),\ 0.448 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.638132516\)
\(L(\frac12)\) \(\approx\) \(1.638132516\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.793 + 0.608i)T \)
7 \( 1 \)
11 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 0.765iT - T^{2} \)
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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

−9.524271584919886338815297326120, −8.615674162226494475662166549856, −7.75380722725464468804547197894, −7.07362061494020660663617212856, −6.19202470148951708455733773580, −5.74151265173976448930610316092, −4.40068230463073569133338789802, −2.97231861802689468288365934256, −2.48632979688723924292027660148, −1.29620974254919121976023908884, 2.02144072343958139596132942005, 2.69400516866452084125244671334, 3.82453088418457046885107099676, 4.58024522000142001569879048354, 5.48349722744867923191820706183, 6.65452882789109225712434752067, 7.63213830079566009313238238885, 8.210516435799149603883368482791, 8.773557737439764596744189923850, 9.751703309670565265782290738936

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