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Properties

Label 1-317-317.110-r0-0-0
Degree $1$
Conductor $317$
Sign $0.814 + 0.579i$
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  + (−0.405 − 0.914i)2-s + (0.293 + 0.955i)3-s + (−0.671 + 0.741i)4-s + (0.804 + 0.594i)5-s + (0.754 − 0.656i)6-s + (0.441 + 0.897i)7-s + (0.949 + 0.312i)8-s + (−0.827 + 0.561i)9-s + (0.216 − 0.976i)10-s + (0.641 − 0.767i)11-s + (−0.905 − 0.423i)12-s + (0.578 − 0.815i)13-s + (0.641 − 0.767i)14-s + (−0.331 + 0.943i)15-s + (−0.0992 − 0.995i)16-s + (0.138 − 0.990i)17-s + ⋯
L(s)  = 1  + (−0.405 − 0.914i)2-s + (0.293 + 0.955i)3-s + (−0.671 + 0.741i)4-s + (0.804 + 0.594i)5-s + (0.754 − 0.656i)6-s + (0.441 + 0.897i)7-s + (0.949 + 0.312i)8-s + (−0.827 + 0.561i)9-s + (0.216 − 0.976i)10-s + (0.641 − 0.767i)11-s + (−0.905 − 0.423i)12-s + (0.578 − 0.815i)13-s + (0.641 − 0.767i)14-s + (−0.331 + 0.943i)15-s + (−0.0992 − 0.995i)16-s + (0.138 − 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.814 + 0.579i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.814 + 0.579i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.262522920 + 0.4034665707i\)
\(L(\frac12)\) \(\approx\) \(1.262522920 + 0.4034665707i\)
\(L(1)\) \(\approx\) \(1.104653040 + 0.1164369862i\)
\(L(1)\) \(\approx\) \(1.104653040 + 0.1164369862i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (-0.405 - 0.914i)T \)
3 \( 1 + (0.293 + 0.955i)T \)
5 \( 1 + (0.804 + 0.594i)T \)
7 \( 1 + (0.441 + 0.897i)T \)
11 \( 1 + (0.641 - 0.767i)T \)
13 \( 1 + (0.578 - 0.815i)T \)
17 \( 1 + (0.138 - 0.990i)T \)
19 \( 1 + (0.754 + 0.656i)T \)
23 \( 1 + (0.0596 + 0.998i)T \)
29 \( 1 + (0.441 - 0.897i)T \)
31 \( 1 + (-0.780 + 0.625i)T \)
37 \( 1 + (-0.476 + 0.878i)T \)
41 \( 1 + (0.368 - 0.929i)T \)
43 \( 1 + (-0.936 - 0.350i)T \)
47 \( 1 + (-0.780 + 0.625i)T \)
53 \( 1 + (0.754 + 0.656i)T \)
59 \( 1 + (-0.936 + 0.350i)T \)
61 \( 1 + (-0.980 + 0.197i)T \)
67 \( 1 + (0.996 - 0.0794i)T \)
71 \( 1 + (-0.331 - 0.943i)T \)
73 \( 1 + (0.511 - 0.859i)T \)
79 \( 1 + (-0.671 - 0.741i)T \)
83 \( 1 + (0.921 - 0.387i)T \)
89 \( 1 + (-0.405 - 0.914i)T \)
97 \( 1 + (-0.476 + 0.878i)T \)
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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

−24.93565580543147251046302605094, −24.3710150465552890238780259886, −23.64803692441568187240296789409, −22.867008924363189826895554015225, −21.526695834004488126886328611, −20.20904933447777648699970374687, −19.770338047519060442636939278799, −18.42958721558659463081616022755, −17.81146902810878859726044742752, −17.03270450014452907170022285550, −16.40872791336556442640012184777, −14.77373666224430910578102991950, −14.19092742235120179243485849831, −13.40279048287677032593073895247, −12.576201375687637675897846679724, −11.10626964189939008015349391803, −9.830006511642992681398036424822, −8.93199710322999370457821281567, −8.13697257902606229576662892475, −6.97229162064070281524958543265, −6.43878682954881929465403711513, −5.16740203793396432178077481069, −4.02906414996175997443562411777, −1.83109207641991978787702943455, −1.12973203923357602191911989178, 1.59655467147427383743995648547, 2.915776401704515394260721808924, 3.457854527168226201745252226633, 5.05661894010545901849792397462, 5.86773140336965000632639424252, 7.75731962091022748142537340515, 8.82414301958827044633910674364, 9.46171989462733709902075935173, 10.367515873115803116993069418888, 11.23847594158791007618978735173, 11.98290318692254075125253039568, 13.60610608760226035558994212472, 14.06562765904926179500322110419, 15.233781434938599700516506917792, 16.29148965401623943758876388928, 17.36472685543031665905124727565, 18.18252135318511707755520510771, 18.952823288215162173923154631044, 20.06720756958300354974604720325, 20.97316697070832365562360156272, 21.50079806673625080843526090494, 22.28371927394881008830133824289, 22.86580743650257867596839848861, 24.88188585721647664712307614635, 25.388994659100866137700376769298

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