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Properties

Label 1-317-317.161-r0-0-0
Degree $1$
Conductor $317$
Sign $0.935 - 0.354i$
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  + (0.0992 − 0.995i)2-s + (−0.368 − 0.929i)3-s + (−0.980 − 0.197i)4-s + (0.827 + 0.561i)5-s + (−0.961 + 0.274i)6-s + (−0.255 + 0.966i)7-s + (−0.293 + 0.955i)8-s + (−0.727 + 0.685i)9-s + (0.641 − 0.767i)10-s + (−0.936 − 0.350i)11-s + (0.177 + 0.984i)12-s + (0.780 + 0.625i)13-s + (0.936 + 0.350i)14-s + (0.216 − 0.976i)15-s + (0.921 + 0.387i)16-s + (−0.848 + 0.528i)17-s + ⋯
L(s)  = 1  + (0.0992 − 0.995i)2-s + (−0.368 − 0.929i)3-s + (−0.980 − 0.197i)4-s + (0.827 + 0.561i)5-s + (−0.961 + 0.274i)6-s + (−0.255 + 0.966i)7-s + (−0.293 + 0.955i)8-s + (−0.727 + 0.685i)9-s + (0.641 − 0.767i)10-s + (−0.936 − 0.350i)11-s + (0.177 + 0.984i)12-s + (0.780 + 0.625i)13-s + (0.936 + 0.350i)14-s + (0.216 − 0.976i)15-s + (0.921 + 0.387i)16-s + (−0.848 + 0.528i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.935 - 0.354i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.935 - 0.354i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9631334863 - 0.1762271998i\)
\(L(\frac12)\) \(\approx\) \(0.9631334863 - 0.1762271998i\)
\(L(1)\) \(\approx\) \(0.8523363749 - 0.3560777973i\)
\(L(1)\) \(\approx\) \(0.8523363749 - 0.3560777973i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (0.0992 - 0.995i)T \)
3 \( 1 + (-0.368 - 0.929i)T \)
5 \( 1 + (0.827 + 0.561i)T \)
7 \( 1 + (-0.255 + 0.966i)T \)
11 \( 1 + (-0.936 - 0.350i)T \)
13 \( 1 + (0.780 + 0.625i)T \)
17 \( 1 + (-0.848 + 0.528i)T \)
19 \( 1 + (0.961 + 0.274i)T \)
23 \( 1 + (0.971 + 0.236i)T \)
29 \( 1 + (0.255 + 0.966i)T \)
31 \( 1 + (-0.905 + 0.423i)T \)
37 \( 1 + (-0.405 - 0.914i)T \)
41 \( 1 + (-0.0596 + 0.998i)T \)
43 \( 1 + (0.138 - 0.990i)T \)
47 \( 1 + (0.905 - 0.423i)T \)
53 \( 1 + (-0.961 - 0.274i)T \)
59 \( 1 + (0.138 + 0.990i)T \)
61 \( 1 + (0.700 + 0.714i)T \)
67 \( 1 + (0.949 + 0.312i)T \)
71 \( 1 + (-0.216 - 0.976i)T \)
73 \( 1 + (-0.545 - 0.838i)T \)
79 \( 1 + (-0.980 + 0.197i)T \)
83 \( 1 + (-0.0198 + 0.999i)T \)
89 \( 1 + (-0.0992 + 0.995i)T \)
97 \( 1 + (0.405 + 0.914i)T \)
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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

−25.37129063488331074043593060678, −24.319679468210237471327958676074, −23.36282256152250821207999616960, −22.69748478210441568135599725551, −21.890779124269680760073766649201, −20.69529498081713430936352606748, −20.37635603431847003066982003991, −18.51031142012240129940537415959, −17.4972626107795225826460620660, −17.15226725458832484840687731070, −15.90981853778675435745620112695, −15.71906966308681999427597860641, −14.31209746165023836023034512917, −13.43625179706846830061812028015, −12.80386325830544927615181068847, −11.083633252883275272836140084043, −10.07115936032576682524361477836, −9.40718363633484651739828412910, −8.37436508114518535348510724443, −7.113708004168902641208798156105, −6.00052080429132376965591388265, −5.12868719097162973447043371190, −4.39558232642175624884963907104, −3.0979709270250389896016382165, −0.680783234158376830963555905130, 1.47287375151105067382290764114, 2.37767074616944303202672118761, 3.277675576327029315272210782087, 5.251745718304568565733238717085, 5.81538986233922193523923088833, 6.98455199962807129843972164928, 8.50819446686269847558894571618, 9.24859303374730445200569902858, 10.644494399619765238823351126979, 11.18168696842933357579175707061, 12.29190432568831261593435166544, 13.14718577906901332574938487994, 13.7122230286947106556668930889, 14.75337626082660364090013593604, 16.18207064952005701355567560681, 17.53619221018906267384530677840, 18.28766795612873493747490127187, 18.63417359043942804784234404800, 19.53064104308414025304595827330, 20.78128178276119843320254588751, 21.68871862575791174404437069001, 22.23957422654064895228648317876, 23.22382637804656683741192497149, 24.03024507131258807243035154809, 25.15912535553939681461448783561

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