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Properties

Label 2-1617-231.131-c0-0-7
Degree $2$
Conductor $1617$
Sign $0.255 - 0.966i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.255 - 0.966i$
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: \((2,\ 1617,\ (\ :0),\ 0.255 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.398910008\)
\(L(\frac12)\) \(\approx\) \(1.398910008\)
\(L(1)\) 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

−9.334503146033162643148271020299, −9.153608677104769561619013442332, −8.395647706783260355859073173853, −7.44632209989365778294412329588, −6.52505136523920863024278166069, −5.52283896345005327226289299960, −4.76572306895437735912646362596, −3.66559687873835367213872962413, −3.16295390616562574437685889908, −1.73283117774167686592339798533, 1.22622638758779006586772051444, 2.20546677995443833444245776402, 3.02238991209742770976326438428, 4.50576901186772689202548047341, 5.73643516103411695315390036550, 6.24372194963860806710379630753, 7.04820778809119396882014946542, 7.41591048879899063805052656250, 8.758559026308897867343965566291, 9.370806470160060997964563079239

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