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Properties

Label 1-317-317.251-r0-0-0
Degree $1$
Conductor $317$
Sign $-0.849 - 0.528i$
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  + (0.293 + 0.955i)2-s + (−0.905 + 0.423i)3-s + (−0.827 + 0.561i)4-s + (0.216 + 0.976i)5-s + (−0.671 − 0.741i)6-s + (0.700 + 0.714i)7-s + (−0.780 − 0.625i)8-s + (0.641 − 0.767i)9-s + (−0.869 + 0.494i)10-s + (−0.476 + 0.878i)11-s + (0.511 − 0.859i)12-s + (0.441 + 0.897i)13-s + (−0.476 + 0.878i)14-s + (−0.610 − 0.792i)15-s + (0.368 − 0.929i)16-s + (−0.0992 + 0.995i)17-s + ⋯
L(s)  = 1  + (0.293 + 0.955i)2-s + (−0.905 + 0.423i)3-s + (−0.827 + 0.561i)4-s + (0.216 + 0.976i)5-s + (−0.671 − 0.741i)6-s + (0.700 + 0.714i)7-s + (−0.780 − 0.625i)8-s + (0.641 − 0.767i)9-s + (−0.869 + 0.494i)10-s + (−0.476 + 0.878i)11-s + (0.511 − 0.859i)12-s + (0.441 + 0.897i)13-s + (−0.476 + 0.878i)14-s + (−0.610 − 0.792i)15-s + (0.368 − 0.929i)16-s + (−0.0992 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $-0.849 - 0.528i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ -0.849 - 0.528i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2601691019 + 0.9109545563i\)
\(L(\frac12)\) \(\approx\) \(-0.2601691019 + 0.9109545563i\)
\(L(1)\) \(\approx\) \(0.4413844143 + 0.7825319962i\)
\(L(1)\) \(\approx\) \(0.4413844143 + 0.7825319962i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (0.293 + 0.955i)T \)
3 \( 1 + (-0.905 + 0.423i)T \)
5 \( 1 + (0.216 + 0.976i)T \)
7 \( 1 + (0.700 + 0.714i)T \)
11 \( 1 + (-0.476 + 0.878i)T \)
13 \( 1 + (0.441 + 0.897i)T \)
17 \( 1 + (-0.0992 + 0.995i)T \)
19 \( 1 + (-0.671 + 0.741i)T \)
23 \( 1 + (0.754 - 0.656i)T \)
29 \( 1 + (0.700 - 0.714i)T \)
31 \( 1 + (-0.255 - 0.966i)T \)
37 \( 1 + (0.949 - 0.312i)T \)
41 \( 1 + (-0.177 - 0.984i)T \)
43 \( 1 + (-0.405 - 0.914i)T \)
47 \( 1 + (-0.255 - 0.966i)T \)
53 \( 1 + (-0.671 + 0.741i)T \)
59 \( 1 + (-0.405 + 0.914i)T \)
61 \( 1 + (-0.727 - 0.685i)T \)
67 \( 1 + (0.578 - 0.815i)T \)
71 \( 1 + (-0.610 + 0.792i)T \)
73 \( 1 + (0.987 + 0.158i)T \)
79 \( 1 + (-0.827 - 0.561i)T \)
83 \( 1 + (0.0596 + 0.998i)T \)
89 \( 1 + (0.293 + 0.955i)T \)
97 \( 1 + (0.949 - 0.312i)T \)
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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

−24.26141272195612532426546396518, −23.570473527266193972007578184803, −23.061343165500689320642672287519, −21.7326523127770950130476925261, −21.20163828631261807497815779820, −20.24201497309714403038329286773, −19.4478310243423691450007916775, −18.163960459200426213782830979179, −17.70937207041227670543491290566, −16.72129153610273741042549976962, −15.709025972154958088744591672208, −14.1366578662375171339221427207, −13.188493078469316985383576810795, −12.87759472377362504243931506983, −11.51424253736122920205668328773, −11.04221539324595599010510201352, −10.081831053969318546604660874856, −8.766697299688455690064962574928, −7.80420266764077656016008398603, −6.21785406712545226870848621940, −5.0488454082326082216119326690, −4.72540546593525085994959359524, −3.060045843657629263372172499051, −1.44525995606626825977166053825, −0.68036408905100059720709498827, 2.115236104532054469779482568821, 3.86346265524809122937772594587, 4.72937878306283032648114717681, 5.8812088264898971157619265512, 6.46644790809117858517099123712, 7.553834127749322887254844555261, 8.76699466409239250134655912679, 9.92579788622428538523572687966, 10.888715197213584590244797902609, 11.953300942172566397320604268681, 12.84303157893146584781870319466, 14.17467033492625983309553071518, 15.12998389201545886827111147978, 15.38404880176362900686588209560, 16.777693065145634132595836742196, 17.39715563285772265543900164253, 18.38362415491439502395576682628, 18.75374367167549985141529413964, 20.96763457631882954690153822417, 21.47986198842621705124003709655, 22.264470058131887819815757160195, 23.179939687433698958750134220528, 23.64255418638331283186005329276, 24.778599006864655787052649358792, 25.74624802380979508139253191036

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