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Properties

Label 1-317-317.144-r0-0-0
Degree $1$
Conductor $317$
Sign $0.379 - 0.925i$
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  + (−0.848 − 0.528i)2-s + (−0.921 − 0.387i)3-s + (0.441 + 0.897i)4-s + (0.980 + 0.197i)5-s + (0.578 + 0.815i)6-s + (−0.905 + 0.423i)7-s + (0.0992 − 0.995i)8-s + (0.700 + 0.714i)9-s + (−0.727 − 0.685i)10-s + (−0.992 − 0.119i)11-s + (−0.0596 − 0.998i)12-s + (−0.293 − 0.955i)13-s + (0.992 + 0.119i)14-s + (−0.827 − 0.561i)15-s + (−0.610 + 0.792i)16-s + (0.331 − 0.943i)17-s + ⋯
L(s)  = 1  + (−0.848 − 0.528i)2-s + (−0.921 − 0.387i)3-s + (0.441 + 0.897i)4-s + (0.980 + 0.197i)5-s + (0.578 + 0.815i)6-s + (−0.905 + 0.423i)7-s + (0.0992 − 0.995i)8-s + (0.700 + 0.714i)9-s + (−0.727 − 0.685i)10-s + (−0.992 − 0.119i)11-s + (−0.0596 − 0.998i)12-s + (−0.293 − 0.955i)13-s + (0.992 + 0.119i)14-s + (−0.827 − 0.561i)15-s + (−0.610 + 0.792i)16-s + (0.331 − 0.943i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4852802377 - 0.3256065284i\)
\(L(\frac12)\) \(\approx\) \(0.4852802377 - 0.3256065284i\)
\(L(1)\) \(\approx\) \(0.5518268039 - 0.1821039082i\)
\(L(1)\) \(\approx\) \(0.5518268039 - 0.1821039082i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (-0.848 - 0.528i)T \)
3 \( 1 + (-0.921 - 0.387i)T \)
5 \( 1 + (0.980 + 0.197i)T \)
7 \( 1 + (-0.905 + 0.423i)T \)
11 \( 1 + (-0.992 - 0.119i)T \)
13 \( 1 + (-0.293 - 0.955i)T \)
17 \( 1 + (0.331 - 0.943i)T \)
19 \( 1 + (-0.578 + 0.815i)T \)
23 \( 1 + (0.996 + 0.0794i)T \)
29 \( 1 + (0.905 + 0.423i)T \)
31 \( 1 + (0.368 - 0.929i)T \)
37 \( 1 + (0.138 + 0.990i)T \)
41 \( 1 + (0.0198 - 0.999i)T \)
43 \( 1 + (0.888 - 0.459i)T \)
47 \( 1 + (-0.368 + 0.929i)T \)
53 \( 1 + (0.578 - 0.815i)T \)
59 \( 1 + (0.888 + 0.459i)T \)
61 \( 1 + (-0.255 - 0.966i)T \)
67 \( 1 + (-0.405 - 0.914i)T \)
71 \( 1 + (0.827 - 0.561i)T \)
73 \( 1 + (0.754 - 0.656i)T \)
79 \( 1 + (0.441 - 0.897i)T \)
83 \( 1 + (-0.869 + 0.494i)T \)
89 \( 1 + (0.848 + 0.528i)T \)
97 \( 1 + (-0.138 - 0.990i)T \)
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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

−25.58727511010670468063133290671, −24.48212562697874215652395442263, −23.52749023324542894881458130776, −23.00918614118461577294096353952, −21.55537263051456136448432108158, −21.13815633098951996440804882222, −19.73793686169325693189680741036, −18.854901709682262143875460477, −17.83506271553772238822654220524, −17.16926351351187685181519267223, −16.49986366502426741299864782550, −15.750894304575121673211411147224, −14.675367239535554573946983219792, −13.39899593277012970054480025491, −12.486600883454293969127997659773, −11.01080938564079955608684723759, −10.29043687369241585936096675910, −9.655466887414193691120143062788, −8.7023514150077879778461392447, −7.06974327270614222969605468776, −6.443778620521727498899384537619, −5.51963558770613621072502335591, −4.516010412732275760550463443139, −2.53375419430158761991066655337, −1.01312621625481422986943908298, 0.70266984004187599517063179800, 2.246146996912073118543508405229, 3.07157612801991313710208677254, 5.10330534147861605316625935577, 6.06625215870647511295589032366, 7.00035452150446973616912952128, 8.07281405082109209183471094297, 9.44937209596675832809925611848, 10.22231908695311502155607423081, 10.81965908076966461806625907224, 12.15009875469417766698680121120, 12.81695567095709123364068482739, 13.53206530796959012474923961137, 15.37640943619662366769222216202, 16.304986276729538743550059888363, 17.08989723767876900421662126536, 17.92968677855682681946726866667, 18.609655430348041072275755140133, 19.22599368489616292192623155857, 20.675620249985248879316582685895, 21.333776496280567719985520789174, 22.39084876117857273438445587869, 22.838211721863357617637297600708, 24.35171832801846768260717943939, 25.34933016843429406929250805990

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