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Properties

Label 1-5241-5241.5240-r0-0-0
Degree $1$
Conductor $5241$
Sign $1$
Analytic cond. $24.3391$
Root an. cond. $24.3391$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5241 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5241 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5241\)    =    \(3 \cdot 1747\)
Sign: $1$
Analytic conductor: \(24.3391\)
Root analytic conductor: \(24.3391\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5241} (5240, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 5241,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.253536826\)
\(L(\frac12)\) \(\approx\) \(4.253536826\)
\(L(1)\) \(\approx\) \(2.227664509\)
\(L(1)\) \(\approx\) \(2.227664509\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
1747 \( 1 \)
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 \( 1 - T \)
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

−17.70881198383319493853119635200, −17.198331100419834657516944168802, −16.59397104525626087049685641198, −15.87690844054797963448457424082, −15.27821527052289436983144672482, −14.41258776041154682294457508840, −13.95317736486632522114919534573, −13.38618182643212085511306068648, −12.75487746529069046323826835840, −12.04014671948622788740813324819, −11.57700729002383477774333791990, −10.49962611820311332142964372491, −9.950082838506073479573382514632, −9.39615972506210710887721056787, −8.58485465388722994339518214773, −7.29361255624994724676624978452, −6.904495287392644065696186137477, −6.20713541083947351938607876353, −5.62859583564405582061301903309, −4.90744494731475789598142654482, −4.01401067683220677870862320353, −3.3940051924703167013653131768, −2.422046696332539119013647260240, −2.04025144247440599431011115088, −0.89201074509445929287607067959, 0.89201074509445929287607067959, 2.04025144247440599431011115088, 2.422046696332539119013647260240, 3.3940051924703167013653131768, 4.01401067683220677870862320353, 4.90744494731475789598142654482, 5.62859583564405582061301903309, 6.20713541083947351938607876353, 6.904495287392644065696186137477, 7.29361255624994724676624978452, 8.58485465388722994339518214773, 9.39615972506210710887721056787, 9.950082838506073479573382514632, 10.49962611820311332142964372491, 11.57700729002383477774333791990, 12.04014671948622788740813324819, 12.75487746529069046323826835840, 13.38618182643212085511306068648, 13.95317736486632522114919534573, 14.41258776041154682294457508840, 15.27821527052289436983144672482, 15.87690844054797963448457424082, 16.59397104525626087049685641198, 17.198331100419834657516944168802, 17.70881198383319493853119635200

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