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Properties

Label 1-249-249.248-r0-0-0
Degree $1$
Conductor $249$
Sign $1$
Analytic cond. $1.15635$
Root an. cond. $1.15635$
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 + 4-s + 5-s + 7-s + 8-s + 10-s − 11-s − 13-s + 14-s + 16-s − 17-s − 19-s + 20-s − 22-s − 23-s + 25-s − 26-s + 28-s − 29-s + 31-s + 32-s − 34-s + 35-s + 37-s − 38-s + 40-s − 41-s + ⋯
L(s)  = 1  + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 11-s − 13-s + 14-s + 16-s − 17-s − 19-s + 20-s − 22-s − 23-s + 25-s − 26-s + 28-s − 29-s + 31-s + 32-s − 34-s + 35-s + 37-s − 38-s + 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 249 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 249 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(249\)    =    \(3 \cdot 83\)
Sign: $1$
Analytic conductor: \(1.15635\)
Root analytic conductor: \(1.15635\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{249} (248, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 249,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.597070678\)
\(L(\frac12)\) \(\approx\) \(2.597070678\)
\(L(1)\) \(\approx\) \(2.110939929\)
\(L(1)\) \(\approx\) \(2.110939929\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
83 \( 1 \)
good2 \( 1 + T \)
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 \( 1 - T \)
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

−25.8687206247431958040623220817, −24.83714989315363496840303413467, −24.22229691114189882771493675201, −23.41777421709149883411302260326, −22.09472585077956116049328232097, −21.61895553345012543317610769510, −20.75446556978928068510530358937, −19.99714310138811456158497720147, −18.55387752285817394382043717320, −17.5183488035436213934821181107, −16.74206594336541431740212199160, −15.36651761521539324086246140447, −14.6922358914888236811020801837, −13.69323470192743623921954155989, −13.011559486202480654098695561869, −11.8929854283077532220833044422, −10.80295643319198079220011192273, −10.02051068018239253343217807606, −8.42213770325342055636915824211, −7.29932118968656081324437475650, −6.115325397128527539952410900374, −5.14781238634519180908805192869, −4.363826409583354637550021413545, −2.54383888288752655654382051326, −1.923683600109434044991571155861, 1.923683600109434044991571155861, 2.54383888288752655654382051326, 4.363826409583354637550021413545, 5.14781238634519180908805192869, 6.115325397128527539952410900374, 7.29932118968656081324437475650, 8.42213770325342055636915824211, 10.02051068018239253343217807606, 10.80295643319198079220011192273, 11.8929854283077532220833044422, 13.011559486202480654098695561869, 13.69323470192743623921954155989, 14.6922358914888236811020801837, 15.36651761521539324086246140447, 16.74206594336541431740212199160, 17.5183488035436213934821181107, 18.55387752285817394382043717320, 19.99714310138811456158497720147, 20.75446556978928068510530358937, 21.61895553345012543317610769510, 22.09472585077956116049328232097, 23.41777421709149883411302260326, 24.22229691114189882771493675201, 24.83714989315363496840303413467, 25.8687206247431958040623220817

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