Properties

Label 1-2617-2617.2616-r0-0-0
Degree $1$
Conductor $2617$
Sign $1$
Analytic cond. $12.1532$
Root an. cond. $12.1532$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 14-s − 15-s + 16-s + 17-s + 18-s − 19-s − 20-s − 21-s − 22-s + 23-s + 24-s + 25-s + 26-s + 27-s − 28-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 14-s − 15-s + 16-s + 17-s + 18-s − 19-s − 20-s − 21-s − 22-s + 23-s + 24-s + 25-s + 26-s + 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2617 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2617 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2617\)
Sign: $1$
Analytic conductor: \(12.1532\)
Root analytic conductor: \(12.1532\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2617} (2616, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2617,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.924908843\)
\(L(\frac12)\) \(\approx\) \(3.924908843\)
\(L(1)\) \(\approx\) \(2.270042235\)
\(L(1)\) \(\approx\) \(2.270042235\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2617 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + T \)
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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

−19.35190800781782717615414244951, −19.00246400194229060716576692351, −18.35341014980478231678153711664, −16.731185970475396267447971330313, −16.23837863089921362060052978463, −15.57033538730991282270605583211, −15.15076719742036402041989363492, −14.43495007392961366394908709371, −13.524111627896359155228098958752, −12.941982245320922137305267492776, −12.583028486274177770051842583882, −11.63661206563993825639933216255, −10.63439008090611473779989910434, −10.24870501042819754933643526242, −9.03578228833945835828602906250, −8.255133055996465231562105904021, −7.6120272340693973127467221468, −6.88725406064618624711545540428, −6.145602913511372966819042643749, −5.04946941445187829558665688970, −4.25975177312680545589948365606, −3.324678948900417196282514992383, −3.227592377657076850661018182099, −2.204521350870269849190107313360, −0.96647912155492345020597284152, 0.96647912155492345020597284152, 2.204521350870269849190107313360, 3.227592377657076850661018182099, 3.324678948900417196282514992383, 4.25975177312680545589948365606, 5.04946941445187829558665688970, 6.145602913511372966819042643749, 6.88725406064618624711545540428, 7.6120272340693973127467221468, 8.255133055996465231562105904021, 9.03578228833945835828602906250, 10.24870501042819754933643526242, 10.63439008090611473779989910434, 11.63661206563993825639933216255, 12.583028486274177770051842583882, 12.941982245320922137305267492776, 13.524111627896359155228098958752, 14.43495007392961366394908709371, 15.15076719742036402041989363492, 15.57033538730991282270605583211, 16.23837863089921362060052978463, 16.731185970475396267447971330313, 18.35341014980478231678153711664, 19.00246400194229060716576692351, 19.35190800781782717615414244951

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