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Properties

Label 1-317-317.105-r0-0-0
Degree $1$
Conductor $317$
Sign $-0.177 - 0.984i$
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  + (−0.216 − 0.976i)2-s + (0.869 + 0.494i)3-s + (−0.905 + 0.423i)4-s + (0.255 − 0.966i)5-s + (0.293 − 0.955i)6-s + (0.0596 − 0.998i)7-s + (0.610 + 0.792i)8-s + (0.511 + 0.859i)9-s + (−0.999 − 0.0397i)10-s + (0.987 − 0.158i)11-s + (−0.996 − 0.0794i)12-s + (−0.921 − 0.387i)13-s + (−0.987 + 0.158i)14-s + (0.700 − 0.714i)15-s + (0.641 − 0.767i)16-s + (0.827 + 0.561i)17-s + ⋯
L(s)  = 1  + (−0.216 − 0.976i)2-s + (0.869 + 0.494i)3-s + (−0.905 + 0.423i)4-s + (0.255 − 0.966i)5-s + (0.293 − 0.955i)6-s + (0.0596 − 0.998i)7-s + (0.610 + 0.792i)8-s + (0.511 + 0.859i)9-s + (−0.999 − 0.0397i)10-s + (0.987 − 0.158i)11-s + (−0.996 − 0.0794i)12-s + (−0.921 − 0.387i)13-s + (−0.987 + 0.158i)14-s + (0.700 − 0.714i)15-s + (0.641 − 0.767i)16-s + (0.827 + 0.561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $-0.177 - 0.984i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ -0.177 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9437663450 - 1.129488770i\)
\(L(\frac12)\) \(\approx\) \(0.9437663450 - 1.129488770i\)
\(L(1)\) \(\approx\) \(1.046010836 - 0.6545243711i\)
\(L(1)\) \(\approx\) \(1.046010836 - 0.6545243711i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (-0.216 - 0.976i)T \)
3 \( 1 + (0.869 + 0.494i)T \)
5 \( 1 + (0.255 - 0.966i)T \)
7 \( 1 + (0.0596 - 0.998i)T \)
11 \( 1 + (0.987 - 0.158i)T \)
13 \( 1 + (-0.921 - 0.387i)T \)
17 \( 1 + (0.827 + 0.561i)T \)
19 \( 1 + (-0.293 - 0.955i)T \)
23 \( 1 + (-0.405 + 0.914i)T \)
29 \( 1 + (-0.0596 - 0.998i)T \)
31 \( 1 + (-0.0198 + 0.999i)T \)
37 \( 1 + (-0.331 - 0.943i)T \)
41 \( 1 + (0.476 - 0.878i)T \)
43 \( 1 + (0.804 + 0.594i)T \)
47 \( 1 + (0.0198 - 0.999i)T \)
53 \( 1 + (0.293 + 0.955i)T \)
59 \( 1 + (0.804 - 0.594i)T \)
61 \( 1 + (-0.177 - 0.984i)T \)
67 \( 1 + (0.848 + 0.528i)T \)
71 \( 1 + (-0.700 - 0.714i)T \)
73 \( 1 + (0.578 + 0.815i)T \)
79 \( 1 + (-0.905 - 0.423i)T \)
83 \( 1 + (-0.936 + 0.350i)T \)
89 \( 1 + (0.216 + 0.976i)T \)
97 \( 1 + (0.331 + 0.943i)T \)
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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

−25.49272845417511373260222699284, −24.70686886179556759405846486980, −24.04907721082717191812116517953, −22.70545193991118258640031386097, −22.16175185468843691261991591823, −21.06181796220013969594104603275, −19.64032385966623053693576055179, −18.803717758889487454615835773560, −18.4460782134704156170819334149, −17.41483413382279450990145121962, −16.31198529079339759965724468705, −15.04767995781504536554706265033, −14.52999064571311891332888315822, −14.122418775647848757271113690168, −12.72082354937981881045209920766, −11.82418724163137404189888963934, −10.03024630240103666191005238211, −9.414472257085918957806147302151, −8.41782147057747832937670457166, −7.42518848829273701877502685857, −6.62425086802779574174403694542, −5.75811423635537532782659079965, −4.20168683867766936141647285633, −2.89330065277612922079688039565, −1.6923662278732232205162468951, 1.03547093700746651537734590433, 2.18495539046998463860909427423, 3.61231384502099629203058331704, 4.27202158309030320475362565753, 5.3069539464557763668192424066, 7.391681803681674986599787541018, 8.29576156063917630994109621288, 9.27517875843976464106563023066, 9.86829103004829715313785386978, 10.80988405454373100996685018389, 12.08301648092707313010356814856, 13.00238271522282367562680549998, 13.84444999285312542413084262979, 14.528199575643237100382821897069, 15.99119477466339466734724916778, 17.13815617045068797904169854155, 17.42698157536796313167119778778, 19.25558274340648260761014614750, 19.71585797218122760928523241396, 20.2822636806525311834948753653, 21.29299596191144410014263715477, 21.75201073476430318913523906, 22.960273189038886679458761791899, 24.0968191002906719552827316122, 25.06643714565070940207799682321

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