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Properties

Label 1-1617-1617.356-r0-0-0
Degree $1$
Conductor $1617$
Sign $-0.616 + 0.786i$
Analytic cond. $7.50931$
Root an. cond. $7.50931$
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  + (0.691 + 0.722i)2-s + (−0.0448 + 0.998i)4-s + (0.858 + 0.512i)5-s + (−0.753 + 0.657i)8-s + (0.222 + 0.974i)10-s + (0.691 + 0.722i)13-s + (−0.995 − 0.0896i)16-s + (0.936 − 0.351i)17-s + (0.809 + 0.587i)19-s + (−0.550 + 0.834i)20-s + (−0.623 + 0.781i)23-s + (0.473 + 0.880i)25-s + (−0.0448 + 0.998i)26-s + (0.963 + 0.266i)29-s + (−0.309 − 0.951i)31-s + (−0.623 − 0.781i)32-s + ⋯
L(s)  = 1  + (0.691 + 0.722i)2-s + (−0.0448 + 0.998i)4-s + (0.858 + 0.512i)5-s + (−0.753 + 0.657i)8-s + (0.222 + 0.974i)10-s + (0.691 + 0.722i)13-s + (−0.995 − 0.0896i)16-s + (0.936 − 0.351i)17-s + (0.809 + 0.587i)19-s + (−0.550 + 0.834i)20-s + (−0.623 + 0.781i)23-s + (0.473 + 0.880i)25-s + (−0.0448 + 0.998i)26-s + (0.963 + 0.266i)29-s + (−0.309 − 0.951i)31-s + (−0.623 − 0.781i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.616 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.616 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.616 + 0.786i$
Analytic conductor: \(7.50931\)
Root analytic conductor: \(7.50931\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1617,\ (0:\ ),\ -0.616 + 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248905297 + 2.566137037i\)
\(L(\frac12)\) \(\approx\) \(1.248905297 + 2.566137037i\)
\(L(1)\) \(\approx\) \(1.412041244 + 1.117140649i\)
\(L(1)\) \(\approx\) \(1.412041244 + 1.117140649i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.691 + 0.722i)T \)
5 \( 1 + (0.858 + 0.512i)T \)
13 \( 1 + (0.691 + 0.722i)T \)
17 \( 1 + (0.936 - 0.351i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (-0.623 + 0.781i)T \)
29 \( 1 + (0.963 + 0.266i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.963 - 0.266i)T \)
41 \( 1 + (0.753 - 0.657i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.983 + 0.178i)T \)
53 \( 1 + (-0.936 - 0.351i)T \)
59 \( 1 + (0.753 + 0.657i)T \)
61 \( 1 + (-0.936 + 0.351i)T \)
67 \( 1 + T \)
71 \( 1 + (0.550 + 0.834i)T \)
73 \( 1 + (-0.983 + 0.178i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.691 + 0.722i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.332721402600983651102228564292, −19.727438306900557888878384492635, −18.73665724216690168944415969433, −18.02899281017216561684158827575, −17.42763303423113200270321879742, −16.24454390374001582611763224232, −15.73670134806383911507547996119, −14.63840532280305286076494959185, −13.999921558471376533540799550269, −13.42306637530006225662174087698, −12.564342227772456167690498399755, −12.15609340246748948636340765214, −11.029486022415818131067855481619, −10.33630649772422704457776547464, −9.72818468565529753972893896164, −8.86993094355305922006820080956, −8.003462003508944442829050552720, −6.63541051844669914320991519024, −5.94196811411810549061513418429, −5.236689211030098237561134878751, −4.51582569688425198426611017074, −3.384999895475310626573541666232, −2.69551560005042197697638453469, −1.57182661390731906301778471085, −0.87598759340655554549556341132, 1.44594336206150318412969989737, 2.50660201766793461033416704132, 3.43183798605788454858004471565, 4.14792086530275735205417691728, 5.53886085056556954230321087763, 5.654291087800800468724205482393, 6.747444872175069439245742466491, 7.34200962404792289318552917221, 8.27281461264005550254505590869, 9.22712907673854294613933811016, 9.92111543583226342239502691770, 10.969295475820477976684558764297, 11.83100557268220623951526497797, 12.50769871243245193692534574176, 13.59678480984371031384866833129, 13.97986402120856374723752261190, 14.47099604411578187529412066927, 15.54637041201896678981894549852, 16.126594615777802940849349904223, 16.92710062813217418992625502283, 17.61277527175920635274223497078, 18.36836364804829861680354425024, 18.95011049901928975081527223267, 20.28343207448002940756683788793, 21.019806530842543928774505008383

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