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Properties

Label 1-317-317.121-r0-0-0
Degree $1$
Conductor $317$
Sign $-0.577 - 0.816i$
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  + (0.441 − 0.897i)2-s + (0.700 − 0.714i)3-s + (−0.610 − 0.792i)4-s + (0.921 − 0.387i)5-s + (−0.331 − 0.943i)6-s + (0.641 + 0.767i)7-s + (−0.980 + 0.197i)8-s + (−0.0198 − 0.999i)9-s + (0.0596 − 0.998i)10-s + (0.971 − 0.236i)11-s + (−0.992 − 0.119i)12-s + (−0.827 − 0.561i)13-s + (0.971 − 0.236i)14-s + (0.368 − 0.929i)15-s + (−0.255 + 0.966i)16-s + (−0.780 + 0.625i)17-s + ⋯
L(s)  = 1  + (0.441 − 0.897i)2-s + (0.700 − 0.714i)3-s + (−0.610 − 0.792i)4-s + (0.921 − 0.387i)5-s + (−0.331 − 0.943i)6-s + (0.641 + 0.767i)7-s + (−0.980 + 0.197i)8-s + (−0.0198 − 0.999i)9-s + (0.0596 − 0.998i)10-s + (0.971 − 0.236i)11-s + (−0.992 − 0.119i)12-s + (−0.827 − 0.561i)13-s + (0.971 − 0.236i)14-s + (0.368 − 0.929i)15-s + (−0.255 + 0.966i)16-s + (−0.780 + 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.018207476 - 1.966997679i\)
\(L(\frac12)\) \(\approx\) \(1.018207476 - 1.966997679i\)
\(L(1)\) \(\approx\) \(1.270988727 - 1.220643101i\)
\(L(1)\) \(\approx\) \(1.270988727 - 1.220643101i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (0.441 - 0.897i)T \)
3 \( 1 + (0.700 - 0.714i)T \)
5 \( 1 + (0.921 - 0.387i)T \)
7 \( 1 + (0.641 + 0.767i)T \)
11 \( 1 + (0.971 - 0.236i)T \)
13 \( 1 + (-0.827 - 0.561i)T \)
17 \( 1 + (-0.780 + 0.625i)T \)
19 \( 1 + (-0.331 + 0.943i)T \)
23 \( 1 + (0.987 - 0.158i)T \)
29 \( 1 + (0.641 - 0.767i)T \)
31 \( 1 + (-0.727 + 0.685i)T \)
37 \( 1 + (-0.961 - 0.274i)T \)
41 \( 1 + (-0.999 + 0.0397i)T \)
43 \( 1 + (0.578 + 0.815i)T \)
47 \( 1 + (-0.727 + 0.685i)T \)
53 \( 1 + (-0.331 + 0.943i)T \)
59 \( 1 + (0.578 - 0.815i)T \)
61 \( 1 + (-0.869 - 0.494i)T \)
67 \( 1 + (-0.671 - 0.741i)T \)
71 \( 1 + (0.368 + 0.929i)T \)
73 \( 1 + (0.138 + 0.990i)T \)
79 \( 1 + (-0.610 + 0.792i)T \)
83 \( 1 + (0.511 + 0.859i)T \)
89 \( 1 + (0.441 - 0.897i)T \)
97 \( 1 + (-0.961 - 0.274i)T \)
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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

−25.50413887377836198815629305181, −24.698280571515008765433263909020, −23.95548149845898090608177264798, −22.56225620253679741435667006066, −22.02341651122824898526680367502, −21.24779469060557416150177828795, −20.37940545731220163284998260409, −19.287895125535903207436478509769, −17.88130469472368197765752111860, −17.156820055772682020906863940861, −16.511485362182819121189631858153, −15.12889660759611023264814477550, −14.6270621419933008464135570878, −13.83786806041378539445832375320, −13.25661747175627171079097538873, −11.62882544343799761703510732528, −10.46042018968613646022817646094, −9.30170695388561721533762842298, −8.80164454189965139375807304057, −7.25089547662905223910304715841, −6.77717324364765802115842261415, −5.099280870657799900880644432999, −4.540755305209721008203975369680, −3.32873776398852142039254986001, −2.05125996038942982344011983522, 1.360164906385683066172168936174, 2.06116449729604946427978234553, 3.09330148772023531933013959418, 4.51676790559772646857775981945, 5.6622817647556207234308923445, 6.565000263045633662522397770877, 8.3268118568293144543353361068, 8.97967400449419318284511814977, 9.85904195344683810143811136769, 11.131197501864853763101871510632, 12.38004372437642370974620873772, 12.66338696893759240022864614835, 13.87585160985322578396589643869, 14.4839871803812915486657179269, 15.23358012408517948982953622096, 17.22737791478754089533124355364, 17.79319752122595902976320935014, 18.810378984445087648951315298854, 19.565827437293631329870071650412, 20.43390995282497731565713071828, 21.26484805813085735835418616450, 21.89577927855074148065059035583, 22.96033958518911787114393221064, 24.296071760261380648851135502905, 24.655468510459787359435614495998

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