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Properties

Label 2-1617-231.164-c0-0-10
Degree $2$
Conductor $1617$
Sign $-0.832 - 0.553i$
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  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s − 0.999·6-s − 8-s + (−0.499 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 − 0.866i)11-s − 13-s − 0.999·15-s + (0.5 + 0.866i)16-s + (−0.499 + 0.866i)18-s + (0.5 + 0.866i)19-s − 0.999·22-s + (−0.500 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s − 0.999·6-s − 8-s + (−0.499 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 − 0.866i)11-s − 13-s − 0.999·15-s + (0.5 + 0.866i)16-s + (−0.499 + 0.866i)18-s + (0.5 + 0.866i)19-s − 0.999·22-s + (−0.500 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.832 - 0.553i$
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: \((2,\ 1617,\ (\ :0),\ -0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8060561877\)
\(L(\frac12)\) \(\approx\) \(0.8060561877\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.184183711327551646145649060819, −8.338490068019160363418706545999, −7.932748783107998785849115123798, −6.73767342947322257079539929568, −6.01461138727796430549632800623, −4.92566712134190430805831260023, −3.62668311505512108156905735745, −2.81402686424918645504267357614, −1.66849467725264650250704312031, −0.69954799717679210461111291224, 2.49709993097971151098225125605, 3.16285706795461451049843885874, 4.24374285879161238281362749116, 5.10978028550747113663078302098, 6.28908313434472923164938686831, 7.23667933566299335625434474809, 7.47483374876324960241435659138, 8.429510783235846122888409036458, 9.273523274845355394213591742851, 9.732401668569254544707327709242

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