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Properties

Label 1-317-317.172-r0-0-0
Degree $1$
Conductor $317$
Sign $0.249 - 0.968i$
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  + (−0.641 + 0.767i)2-s + (0.476 − 0.878i)3-s + (−0.177 − 0.984i)4-s + (−0.511 − 0.859i)5-s + (0.368 + 0.929i)6-s + (0.971 + 0.236i)7-s + (0.869 + 0.494i)8-s + (−0.545 − 0.838i)9-s + (0.987 + 0.158i)10-s + (0.804 − 0.594i)11-s + (−0.949 − 0.312i)12-s + (0.0198 − 0.999i)13-s + (−0.804 + 0.594i)14-s + (−0.999 + 0.0397i)15-s + (−0.936 + 0.350i)16-s + (0.727 − 0.685i)17-s + ⋯
L(s)  = 1  + (−0.641 + 0.767i)2-s + (0.476 − 0.878i)3-s + (−0.177 − 0.984i)4-s + (−0.511 − 0.859i)5-s + (0.368 + 0.929i)6-s + (0.971 + 0.236i)7-s + (0.869 + 0.494i)8-s + (−0.545 − 0.838i)9-s + (0.987 + 0.158i)10-s + (0.804 − 0.594i)11-s + (−0.949 − 0.312i)12-s + (0.0198 − 0.999i)13-s + (−0.804 + 0.594i)14-s + (−0.999 + 0.0397i)15-s + (−0.936 + 0.350i)16-s + (0.727 − 0.685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.249 - 0.968i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.249 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8230533789 - 0.6375999273i\)
\(L(\frac12)\) \(\approx\) \(0.8230533789 - 0.6375999273i\)
\(L(1)\) \(\approx\) \(0.8872316442 - 0.2432314759i\)
\(L(1)\) \(\approx\) \(0.8872316442 - 0.2432314759i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (-0.641 + 0.767i)T \)
3 \( 1 + (0.476 - 0.878i)T \)
5 \( 1 + (-0.511 - 0.859i)T \)
7 \( 1 + (0.971 + 0.236i)T \)
11 \( 1 + (0.804 - 0.594i)T \)
13 \( 1 + (0.0198 - 0.999i)T \)
17 \( 1 + (0.727 - 0.685i)T \)
19 \( 1 + (-0.368 + 0.929i)T \)
23 \( 1 + (-0.0992 + 0.995i)T \)
29 \( 1 + (-0.971 + 0.236i)T \)
31 \( 1 + (0.996 + 0.0794i)T \)
37 \( 1 + (0.216 - 0.976i)T \)
41 \( 1 + (0.405 - 0.914i)T \)
43 \( 1 + (-0.827 + 0.561i)T \)
47 \( 1 + (-0.996 - 0.0794i)T \)
53 \( 1 + (0.368 - 0.929i)T \)
59 \( 1 + (-0.827 - 0.561i)T \)
61 \( 1 + (0.754 - 0.656i)T \)
67 \( 1 + (-0.610 + 0.792i)T \)
71 \( 1 + (0.999 + 0.0397i)T \)
73 \( 1 + (-0.780 - 0.625i)T \)
79 \( 1 + (-0.177 + 0.984i)T \)
83 \( 1 + (0.138 - 0.990i)T \)
89 \( 1 + (0.641 - 0.767i)T \)
97 \( 1 + (-0.216 + 0.976i)T \)
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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

−25.96087577034520849972818060042, −24.69764132217453581649985031284, −23.3408367332891390274995013628, −22.32274154298086779216031022991, −21.62129585935827467514054445459, −20.85366226378520159817237119781, −19.95468185701561966380492981611, −19.25487539816183540265386282484, −18.3719641453950298589822839176, −17.20351237257858985163116882881, −16.58296019259076043621327806199, −15.17722260829089865086121981520, −14.57459827086387066951258692254, −13.60496793979248928816993294662, −11.97573648340196110374949674946, −11.31948161463874973785972105761, −10.522649574494211672029178484949, −9.6575336106646661971415110175, −8.60102649966381550612968669568, −7.81107271650824900070139173797, −6.69861328665919163918950025979, −4.53333678086564658636306934672, −4.06197217542836051433652728296, −2.83134422406423694469671356510, −1.73041413691156684062161296743, 0.85968446840411577881351758079, 1.757308498958281152638128944738, 3.637486566400994364894617394618, 5.18019915397388454935960353948, 5.974053531644369789620040830453, 7.41975222669242161124742538066, 8.03410248033265781159405179064, 8.67094510606410656715636782870, 9.640007531158441669052116699658, 11.23961630075016128518088649801, 12.047193727438249716132980708891, 13.21969417050623319707249149602, 14.259542927237882834961099353929, 14.901900473145083632760111035794, 15.990114859988702280354957690593, 17.023777188707841869262205351105, 17.71336357944024572719724305882, 18.66626516827375349085221111947, 19.43873179140625028656568543620, 20.24041836639083653438305721475, 21.069453801190493594431240510683, 22.84473262557563165148013751609, 23.52502540487867981691771933366, 24.5560123460351347466616263337, 24.73895720446635889565111801675

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