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Properties

Label 2-1617-231.131-c0-0-4
Degree $2$
Conductor $1617$
Sign $0.949 - 0.315i$
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  + (−0.130 − 0.991i)3-s + (0.5 + 0.866i)4-s + (−0.382 + 0.662i)5-s + (−0.965 + 0.258i)9-s + (0.866 − 0.5i)11-s + (0.793 − 0.608i)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 + (−0.608 − 0.793i)33-s + (−0.707 − 0.707i)36-s + ⋯
L(s)  = 1  + (−0.130 − 0.991i)3-s + (0.5 + 0.866i)4-s + (−0.382 + 0.662i)5-s + (−0.965 + 0.258i)9-s + (0.866 − 0.5i)11-s + (0.793 − 0.608i)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 + (−0.608 − 0.793i)33-s + (−0.707 − 0.707i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.949 - 0.315i$
Analytic conductor: \(0.806988\)
Root analytic conductor: \(0.898325\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :0),\ 0.949 - 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.144879227\)
\(L(\frac12)\) \(\approx\) \(1.144879227\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.130 + 0.991i)T \)
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

−9.418237907210950786910214774169, −8.620139869618451222714473064044, −7.81355895402774016408615761883, −7.28451399485881781598097453863, −6.54707398838101666088070782978, −5.97771694296334941190070923223, −4.53492907222246278311292184331, −3.29723064646551862280348788711, −2.82399894094185606983537339231, −1.43751793996600874649627330818, 1.05715151844119497262999182980, 2.56708747038571780769644029275, 3.76796323658200072354473415579, 4.82732055906421690200052846206, 5.09199304428466524233927147113, 6.38835149331794963673550660222, 6.83260972288610175504944593801, 8.254664278885896461405259102565, 8.863096297615589692258108523327, 9.679023065733363924391773569569

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