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Properties

Label 1-317-317.63-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: $\chi_{317} (63, \cdot )$
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

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

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