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Properties

Label 1-5239-5239.196-r0-0-0
Degree $1$
Conductor $5239$
Sign $0.244 - 0.969i$
Analytic cond. $24.3298$
Root an. cond. $24.3298$
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  + (−0.861 − 0.506i)2-s + (0.541 + 0.840i)3-s + (0.485 + 0.873i)4-s + (0.278 − 0.960i)5-s + (−0.0402 − 0.999i)6-s + (0.184 + 0.982i)7-s + (0.0241 − 0.999i)8-s + (−0.414 + 0.910i)9-s + (−0.726 + 0.686i)10-s + (−0.581 + 0.813i)11-s + (−0.471 + 0.881i)12-s + (0.339 − 0.940i)14-s + (0.958 − 0.285i)15-s + (−0.527 + 0.849i)16-s + (−0.877 + 0.478i)17-s + (0.818 − 0.574i)18-s + ⋯
L(s)  = 1  + (−0.861 − 0.506i)2-s + (0.541 + 0.840i)3-s + (0.485 + 0.873i)4-s + (0.278 − 0.960i)5-s + (−0.0402 − 0.999i)6-s + (0.184 + 0.982i)7-s + (0.0241 − 0.999i)8-s + (−0.414 + 0.910i)9-s + (−0.726 + 0.686i)10-s + (−0.581 + 0.813i)11-s + (−0.471 + 0.881i)12-s + (0.339 − 0.940i)14-s + (0.958 − 0.285i)15-s + (−0.527 + 0.849i)16-s + (−0.877 + 0.478i)17-s + (0.818 − 0.574i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5239\)    =    \(13^{2} \cdot 31\)
Sign: $0.244 - 0.969i$
Analytic conductor: \(24.3298\)
Root analytic conductor: \(24.3298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5239} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5239,\ (0:\ ),\ 0.244 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5774147214 - 0.4501142066i\)
\(L(\frac12)\) \(\approx\) \(0.5774147214 - 0.4501142066i\)
\(L(1)\) \(\approx\) \(0.7589616504 + 0.01941525695i\)
\(L(1)\) \(\approx\) \(0.7589616504 + 0.01941525695i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.861 - 0.506i)T \)
3 \( 1 + (0.541 + 0.840i)T \)
5 \( 1 + (0.278 - 0.960i)T \)
7 \( 1 + (0.184 + 0.982i)T \)
11 \( 1 + (-0.581 + 0.813i)T \)
17 \( 1 + (-0.877 + 0.478i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.861 - 0.506i)T \)
37 \( 1 + (-0.0402 - 0.999i)T \)
41 \( 1 + (0.152 + 0.988i)T \)
43 \( 1 + (0.247 - 0.968i)T \)
47 \( 1 + (0.981 + 0.192i)T \)
53 \( 1 + (0.853 + 0.520i)T \)
59 \( 1 + (0.899 - 0.435i)T \)
61 \( 1 + (0.568 - 0.822i)T \)
67 \( 1 + (-0.919 - 0.391i)T \)
71 \( 1 + (-0.704 - 0.709i)T \)
73 \( 1 + (-0.581 + 0.813i)T \)
79 \( 1 + (-0.999 + 0.0161i)T \)
83 \( 1 + (0.541 - 0.840i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.926 - 0.377i)T \)
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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

−17.971267092515592457343670938539, −17.81486577629994434453805957917, −16.90678266477935923476236806235, −16.18353495673802325666762921378, −15.48619573838281737643200040876, −14.67223818528433517002250250635, −14.18127227541817547848820669030, −13.61048512683106594761227912480, −13.181851857946397625094005260, −11.74174830188746331507470656244, −11.39747529628054802723036588249, −10.510880527538419337576884807045, −10.06641379217294285032402649088, −9.22522841554390055104553180075, −8.48304131588349696386011845278, −7.74056047065286436876613649304, −7.3033023417297904635607853437, −6.810676204991938255448087293884, −5.947493092809503783632522896543, −5.47251731392495936560800048536, −4.05302086490281545751309956192, −3.20311187921095346995999141859, −2.466350161374096079513760959207, −1.689946033474711771151450781999, −0.888918121173858747345976818846, 0.25578287774568861916101679217, 1.72537682337344314397404614524, 2.235246290667695055909545606601, 2.77876534818909651778474444858, 3.97468155647938228036546674608, 4.47340925356852279083738778757, 5.318998242717256699163523168613, 6.026599798942917516073430568810, 7.26685106214135983725442372195, 7.95533533931769893764001932316, 8.6441268338886586991729743108, 9.04608265389800835912951148070, 9.62782356445410066089646443277, 10.21719593868406490873664360475, 11.00978543097385199790594834648, 11.706151025393341105204339377771, 12.43235083334268456265789244948, 13.03226331654957278226763877771, 13.671668815140950021781367002395, 14.762187627006938698287454287320, 15.465109425527269140476320281216, 15.85641343416071726842261745461, 16.4395688764595838542133225346, 17.32393435011666459251428210608, 17.75720506526470581692986719681

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