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Properties

Label 1-317-317.28-r0-0-0
Degree $1$
Conductor $317$
Sign $-0.100 + 0.994i$
Analytic cond. $1.47214$
Root an. cond. $1.47214$
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  + (0.511 + 0.859i)2-s + (0.987 − 0.158i)3-s + (−0.476 + 0.878i)4-s + (0.996 − 0.0794i)5-s + (0.641 + 0.767i)6-s + (0.138 + 0.990i)7-s + (−0.999 + 0.0397i)8-s + (0.949 − 0.312i)9-s + (0.578 + 0.815i)10-s + (−0.780 + 0.625i)11-s + (−0.331 + 0.943i)12-s + (−0.992 − 0.119i)13-s + (−0.780 + 0.625i)14-s + (0.971 − 0.236i)15-s + (−0.545 − 0.838i)16-s + (−0.177 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.511 + 0.859i)2-s + (0.987 − 0.158i)3-s + (−0.476 + 0.878i)4-s + (0.996 − 0.0794i)5-s + (0.641 + 0.767i)6-s + (0.138 + 0.990i)7-s + (−0.999 + 0.0397i)8-s + (0.949 − 0.312i)9-s + (0.578 + 0.815i)10-s + (−0.780 + 0.625i)11-s + (−0.331 + 0.943i)12-s + (−0.992 − 0.119i)13-s + (−0.780 + 0.625i)14-s + (0.971 − 0.236i)15-s + (−0.545 − 0.838i)16-s + (−0.177 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $-0.100 + 0.994i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ -0.100 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.586876427 + 1.754723866i\)
\(L(\frac12)\) \(\approx\) \(1.586876427 + 1.754723866i\)
\(L(1)\) \(\approx\) \(1.581689873 + 1.011080207i\)
\(L(1)\) \(\approx\) \(1.581689873 + 1.011080207i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (0.511 + 0.859i)T \)
3 \( 1 + (0.987 - 0.158i)T \)
5 \( 1 + (0.996 - 0.0794i)T \)
7 \( 1 + (0.138 + 0.990i)T \)
11 \( 1 + (-0.780 + 0.625i)T \)
13 \( 1 + (-0.992 - 0.119i)T \)
17 \( 1 + (-0.177 + 0.984i)T \)
19 \( 1 + (0.641 - 0.767i)T \)
23 \( 1 + (-0.827 - 0.561i)T \)
29 \( 1 + (0.138 - 0.990i)T \)
31 \( 1 + (0.888 + 0.459i)T \)
37 \( 1 + (-0.255 - 0.966i)T \)
41 \( 1 + (0.804 - 0.594i)T \)
43 \( 1 + (-0.905 + 0.423i)T \)
47 \( 1 + (0.888 + 0.459i)T \)
53 \( 1 + (0.641 - 0.767i)T \)
59 \( 1 + (-0.905 - 0.423i)T \)
61 \( 1 + (-0.405 + 0.914i)T \)
67 \( 1 + (0.700 + 0.714i)T \)
71 \( 1 + (0.971 + 0.236i)T \)
73 \( 1 + (-0.610 - 0.792i)T \)
79 \( 1 + (-0.476 - 0.878i)T \)
83 \( 1 + (-0.671 - 0.741i)T \)
89 \( 1 + (0.511 + 0.859i)T \)
97 \( 1 + (-0.255 - 0.966i)T \)
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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

−24.70832285299174358443623792580, −24.18167312100495815012059151314, −22.99138748178602084777344306058, −21.89547107843402767743454314433, −21.34171403449061032648822687163, −20.384402917968605674014875844985, −20.01304339644903403311760454408, −18.75173432593948022314226631719, −18.12456123183840756408638952452, −16.81538051685593054835694750385, −15.59817302257794438251732009853, −14.377398875615031639656386000397, −13.83450858184911411144065913718, −13.37876818135635165373086376362, −12.186940426564510944465425027617, −10.80666406958129274729664311025, −9.98539142935721788621770047599, −9.50191508542309904875244622115, −8.131482068641957043128795319134, −6.93749536147575155268943434729, −5.45170780129801387094282562473, −4.54013710581215144157628951043, −3.27653327996223468926330797052, −2.472065450133559435759774246366, −1.30037014304750586797578784548, 2.1885367249323346193817577049, 2.76283524113019155617719373144, 4.41070614513368803014071419570, 5.357390362851494012820997556467, 6.39830341313041401665958431611, 7.51282377135354643745290035609, 8.4230574359743907640550462775, 9.30239465858883548301497958842, 10.14533996565523978686149797585, 12.18041930990706993620200285654, 12.79596176513880846230349226710, 13.67025821723820468905861974212, 14.54584310328767435346436299738, 15.23054307352427706804011972716, 16.010633307633185858504334301937, 17.514593926792421319444047044685, 17.91654342346319363208345155345, 18.9884294136217007586376396259, 20.24970364025369369895188437920, 21.293970855913068719974439399303, 21.6850446955428008586590203170, 22.678175290529457947962050230556, 24.12842328232349235734242733550, 24.58222092869189957756431706627, 25.264826142673122901753073307341

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