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Properties

Label 1-317-317.156-r0-0-0
Degree $1$
Conductor $317$
Sign $0.512 - 0.858i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.512 - 0.858i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.512 - 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07758052973 - 0.04403546919i\)
\(L(\frac12)\) \(\approx\) \(0.07758052973 - 0.04403546919i\)
\(L(1)\) \(\approx\) \(0.4752024351 + 0.3681834476i\)
\(L(1)\) \(\approx\) \(0.4752024351 + 0.3681834476i\)

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

−25.71175838383517279388724279013, −24.00744511565836671842804374094, −23.378320686387100748526526428650, −22.930816987163975043104784785434, −21.623725225367919661098767310569, −20.53315490934544681558221735877, −19.8671507638680399915757222099, −19.001845918134508145110464653473, −18.64864632006173599594741558759, −17.44477534307119245429107718825, −16.58798555848518760301164807048, −14.79208100683246689246172505457, −14.34527824752500372368854404901, −12.98657070736021785416900115214, −12.63965055773543570982238048286, −11.45686767423599378992922441808, −10.6427965595713460264937215252, −9.65105722294829116803660395406, −8.25799880000087202622965137049, −7.59769431987582771165079919765, −6.65357551115092025847209445363, −4.84016611747189028841950760734, −3.6359052362651161057221046739, −2.81354096450730462142283476800, −1.55461413287177762988013483044, 0.05630299136438841855125116953, 2.76587766614246016793454311068, 3.90231940466461546067264172299, 5.19437565201794616142756304522, 5.466098860114648248004534219361, 7.30896937164310153813549970888, 8.2108545612900063873510325064, 8.90631953506836226928020334954, 9.783834589825975011763976237731, 10.93725852366909561672747092066, 12.360258821675507162400420987434, 13.18430314591068751482843719931, 14.55537185429739808452369905788, 15.257106321985554327665169205041, 15.8283359089919980168573798314, 16.549519267486126861640786789472, 17.52183559273081405627665289692, 18.93738498213256785655908628173, 19.35640908263645415427063985600, 20.732042376376097219306480858267, 21.518099668738904862932412367777, 22.53551563296300552844669147877, 23.24650396047412150674626727231, 24.29518422721988443678156751786, 25.149380245718713969908002932394

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