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Properties

Label 1-317-317.212-r0-0-0
Degree $1$
Conductor $317$
Sign $-0.495 + 0.868i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $-0.495 + 0.868i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ -0.495 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4259316500 + 0.7334102827i\)
\(L(\frac12)\) \(\approx\) \(0.4259316500 + 0.7334102827i\)
\(L(1)\) \(\approx\) \(0.6908567907 + 0.4413513966i\)
\(L(1)\) \(\approx\) \(0.6908567907 + 0.4413513966i\)

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

−24.41182582975613623796858094951, −23.98678101971132392137431542800, −22.639020893422792747450654998323, −22.2880842318218220333828279745, −21.28465795543481859769681730019, −20.5839344727848504711931625264, −19.70368084105600087542106566306, −18.60925082744271525127147661035, −17.6863594119375865212692220304, −16.94332764097365772923665898721, −15.65200240613868464548031464312, −15.08420573681082207648036365552, −13.50914786454293444853208991382, −12.63100998917303377331059236850, −11.81058622647649589969133301760, −11.31554413732063604386871578847, −10.08661795370865819446443044938, −9.05069196986431670052043678803, −8.53379789920394328053710172367, −6.34662229194870717637869991885, −5.485460065202474245545811547028, −4.502815531289736667668495078172, −3.74435588637680455723831431269, −2.02961848153603025066833279838, −0.66931258502137455681075431972, 1.29671466053142330490940339858, 3.51448513613923056539992489355, 4.34104191013645454857833447755, 5.744433079585273693517821351741, 6.68809103838733227833059368270, 7.1137035313601616682790088895, 8.16597467504587122312322037427, 9.617406346195432220922917813161, 10.84028093321575993965384244052, 11.58396987747237310485180584761, 12.76810330336034625060751035546, 14.01296072530078695904978804758, 14.167888739329339003752753617634, 15.78594121232547944094960450626, 16.33661230471910654265869880843, 17.39516668054272603716376188338, 18.00817986409196303900439731788, 18.84421198458229443262954617563, 19.87149235087941491938021031148, 21.49667332916843989570079154704, 22.29995928500877573533173262969, 23.08647146478817189738148973654, 23.46014863081087177809081284171, 24.46854646270346252171016422144, 25.341613805423697932427396718382

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