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Properties

Label 1-317-317.24-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: $\chi_{317} (24, \cdot )$
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

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

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