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Properties

Label 1-317-317.15-r0-0-0
Degree $1$
Conductor $317$
Sign $-0.178 - 0.983i$
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  + (−0.936 − 0.350i)2-s + (−0.405 − 0.914i)3-s + (0.754 + 0.656i)4-s + (−0.545 + 0.838i)5-s + (0.0596 + 0.998i)6-s + (0.578 − 0.815i)7-s + (−0.476 − 0.878i)8-s + (−0.671 + 0.741i)9-s + (0.804 − 0.594i)10-s + (−0.827 + 0.561i)11-s + (0.293 − 0.955i)12-s + (0.996 − 0.0794i)13-s + (−0.827 + 0.561i)14-s + (0.987 + 0.158i)15-s + (0.138 + 0.990i)16-s + (−0.992 + 0.119i)17-s + ⋯
L(s)  = 1  + (−0.936 − 0.350i)2-s + (−0.405 − 0.914i)3-s + (0.754 + 0.656i)4-s + (−0.545 + 0.838i)5-s + (0.0596 + 0.998i)6-s + (0.578 − 0.815i)7-s + (−0.476 − 0.878i)8-s + (−0.671 + 0.741i)9-s + (0.804 − 0.594i)10-s + (−0.827 + 0.561i)11-s + (0.293 − 0.955i)12-s + (0.996 − 0.0794i)13-s + (−0.827 + 0.561i)14-s + (0.987 + 0.158i)15-s + (0.138 + 0.990i)16-s + (−0.992 + 0.119i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $-0.178 - 0.983i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ -0.178 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3664180415 - 0.4387884564i\)
\(L(\frac12)\) \(\approx\) \(0.3664180415 - 0.4387884564i\)
\(L(1)\) \(\approx\) \(0.5307860819 - 0.2406286327i\)
\(L(1)\) \(\approx\) \(0.5307860819 - 0.2406286327i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (-0.936 - 0.350i)T \)
3 \( 1 + (-0.405 - 0.914i)T \)
5 \( 1 + (-0.545 + 0.838i)T \)
7 \( 1 + (0.578 - 0.815i)T \)
11 \( 1 + (-0.827 + 0.561i)T \)
13 \( 1 + (0.996 - 0.0794i)T \)
17 \( 1 + (-0.992 + 0.119i)T \)
19 \( 1 + (0.0596 - 0.998i)T \)
23 \( 1 + (0.921 - 0.387i)T \)
29 \( 1 + (0.578 + 0.815i)T \)
31 \( 1 + (0.949 - 0.312i)T \)
37 \( 1 + (0.641 - 0.767i)T \)
41 \( 1 + (-0.0992 - 0.995i)T \)
43 \( 1 + (-0.727 + 0.685i)T \)
47 \( 1 + (0.949 - 0.312i)T \)
53 \( 1 + (0.0596 - 0.998i)T \)
59 \( 1 + (-0.727 - 0.685i)T \)
61 \( 1 + (-0.961 + 0.274i)T \)
67 \( 1 + (-0.869 - 0.494i)T \)
71 \( 1 + (0.987 - 0.158i)T \)
73 \( 1 + (-0.905 - 0.423i)T \)
79 \( 1 + (0.754 - 0.656i)T \)
83 \( 1 + (0.848 - 0.528i)T \)
89 \( 1 + (-0.936 - 0.350i)T \)
97 \( 1 + (0.641 - 0.767i)T \)
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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

−25.42231189698464243503332308578, −24.68476179344031853976248986974, −23.68306275380833756931051187398, −23.06383331467237293923767835294, −21.47754786018211022990471013668, −20.90838574955655062359529152736, −20.18410246629097931708290306357, −18.94532699585675800640734315449, −18.15139989459453803880955989094, −17.15577394133423476467934141085, −16.34584131173303391778066501564, −15.51995526180803834696224331404, −15.23120496481290213658051611807, −13.66177189247677155464461483757, −12.07895818919308720920898661961, −11.35618798760984635681851648186, −10.59298164427585827414468165936, −9.34874679352244690479716331147, −8.57247566715770940983465616794, −8.01424076236541301504373836405, −6.25281376349583485022471918646, −5.4338807414798696206774275389, −4.48513207503088406087586416060, −2.887909520501771295665693379290, −1.154119772454780911603763222002, 0.61958149931023079282794727993, 2.020840046248818606734755480224, 3.07594721824162468790912685858, 4.60463497710230215640309880831, 6.42668977693343250410328696073, 7.11833799503958569524117218924, 7.844137378994586282197661996384, 8.76243804136469868249983719706, 10.53911433329642279134399530067, 10.883384104928816865426822822870, 11.66430130410472545572163827858, 12.866073073005794236415150692671, 13.69220797006584593947122777479, 15.11741036546590277910279598402, 16.01544660333279552986978518059, 17.20882332092826603130641102959, 17.97526793253384846131118449224, 18.38917292010174865584236061003, 19.47672026120544167443205137328, 20.10676293094784356296434241340, 21.12516533568676385853369745368, 22.375238965085231018663156256707, 23.32851193208775245171467068208, 23.94442504368152560891584148191, 25.05677569891158393794112056671

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