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Properties

Label 1-317-317.135-r0-0-0
Degree $1$
Conductor $317$
Sign $-0.906 + 0.421i$
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  + (0.671 + 0.741i)2-s + (0.827 + 0.561i)3-s + (−0.0992 + 0.995i)4-s + (−0.293 + 0.955i)5-s + (0.138 + 0.990i)6-s + (−0.610 − 0.792i)7-s + (−0.804 + 0.594i)8-s + (0.368 + 0.929i)9-s + (−0.905 + 0.423i)10-s + (−0.177 + 0.984i)11-s + (−0.641 + 0.767i)12-s + (0.331 − 0.943i)13-s + (0.177 − 0.984i)14-s + (−0.780 + 0.625i)15-s + (−0.980 − 0.197i)16-s + (0.961 − 0.274i)17-s + ⋯
L(s)  = 1  + (0.671 + 0.741i)2-s + (0.827 + 0.561i)3-s + (−0.0992 + 0.995i)4-s + (−0.293 + 0.955i)5-s + (0.138 + 0.990i)6-s + (−0.610 − 0.792i)7-s + (−0.804 + 0.594i)8-s + (0.368 + 0.929i)9-s + (−0.905 + 0.423i)10-s + (−0.177 + 0.984i)11-s + (−0.641 + 0.767i)12-s + (0.331 − 0.943i)13-s + (0.177 − 0.984i)14-s + (−0.780 + 0.625i)15-s + (−0.980 − 0.197i)16-s + (0.961 − 0.274i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $-0.906 + 0.421i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ -0.906 + 0.421i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4198723975 + 1.899933794i\)
\(L(\frac12)\) \(\approx\) \(0.4198723975 + 1.899933794i\)
\(L(1)\) \(\approx\) \(1.072844536 + 1.217695555i\)
\(L(1)\) \(\approx\) \(1.072844536 + 1.217695555i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (0.671 + 0.741i)T \)
3 \( 1 + (0.827 + 0.561i)T \)
5 \( 1 + (-0.293 + 0.955i)T \)
7 \( 1 + (-0.610 - 0.792i)T \)
11 \( 1 + (-0.177 + 0.984i)T \)
13 \( 1 + (0.331 - 0.943i)T \)
17 \( 1 + (0.961 - 0.274i)T \)
19 \( 1 + (-0.138 + 0.990i)T \)
23 \( 1 + (-0.992 - 0.119i)T \)
29 \( 1 + (0.610 - 0.792i)T \)
31 \( 1 + (0.216 + 0.976i)T \)
37 \( 1 + (-0.545 + 0.838i)T \)
41 \( 1 + (0.727 - 0.685i)T \)
43 \( 1 + (0.754 - 0.656i)T \)
47 \( 1 + (-0.216 - 0.976i)T \)
53 \( 1 + (0.138 - 0.990i)T \)
59 \( 1 + (0.754 + 0.656i)T \)
61 \( 1 + (0.921 + 0.387i)T \)
67 \( 1 + (0.987 + 0.158i)T \)
71 \( 1 + (0.780 + 0.625i)T \)
73 \( 1 + (-0.476 + 0.878i)T \)
79 \( 1 + (-0.0992 - 0.995i)T \)
83 \( 1 + (0.700 + 0.714i)T \)
89 \( 1 + (-0.671 - 0.741i)T \)
97 \( 1 + (0.545 - 0.838i)T \)
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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

−24.42686764162584840300826949264, −24.02765564802326201055096927189, −23.18743908259611518681544445223, −21.733751532234314188430647161192, −21.2650763586676768870567442601, −20.32358982641661552740495329283, −19.230139153221375278395236494928, −19.17163156644014931403671384124, −17.972167262793045074714700704162, −16.26354944935145665050646609929, −15.63457622785292173809616344364, −14.4236960955835395283473908367, −13.63683266648958198238804393007, −12.785023056383370079433397992606, −12.17926489841137305864597733710, −11.23875784935127508837345880010, −9.603878016260886160517495768048, −9.02763733535065310216354319116, −8.07564926822666702315082052617, −6.47302186260952351752888799995, −5.60007388146680422115976583997, −4.20938244695909801592624838251, −3.26606320514834017348197658587, −2.20129887023944701558356015393, −0.94417177951990324355957738961, 2.48931821681122880147579611922, 3.51271548673760478042842844799, 4.06674703958038863806739543398, 5.46152840500361019376198861233, 6.76107263059326510196865168626, 7.60211036172418447452866706403, 8.284803956342897383690335947874, 9.97778818298100849188274010931, 10.34969004543527581573471532204, 11.98942348488726785248193428300, 13.01936323601621499191212305017, 14.05554961619359773369824909599, 14.52669228590690510140947894918, 15.57998559003541775738260585944, 16.02945354188854002911728253930, 17.24460578656336272344854022242, 18.30239923107567508273567861951, 19.43522239828052968975905530037, 20.42595019145230378566976192038, 21.07930912892780925340825448795, 22.36056287470338097385734894137, 22.8091346756888784164547610625, 23.52313660318516430307770162678, 24.972731973872727728292178830086, 25.66507671000206599019302203523

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