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Properties

Label 1-317-317.124-r0-0-0
Degree $1$
Conductor $317$
Sign $0.348 - 0.937i$
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  + (0.780 − 0.625i)2-s + (0.255 + 0.966i)3-s + (0.216 − 0.976i)4-s + (0.610 − 0.792i)5-s + (0.804 + 0.594i)6-s + (−0.727 − 0.685i)7-s + (−0.441 − 0.897i)8-s + (−0.869 + 0.494i)9-s + (−0.0198 − 0.999i)10-s + (0.996 + 0.0794i)11-s + (0.999 − 0.0397i)12-s + (0.980 − 0.197i)13-s + (−0.996 − 0.0794i)14-s + (0.921 + 0.387i)15-s + (−0.905 − 0.423i)16-s + (−0.293 − 0.955i)17-s + ⋯
L(s)  = 1  + (0.780 − 0.625i)2-s + (0.255 + 0.966i)3-s + (0.216 − 0.976i)4-s + (0.610 − 0.792i)5-s + (0.804 + 0.594i)6-s + (−0.727 − 0.685i)7-s + (−0.441 − 0.897i)8-s + (−0.869 + 0.494i)9-s + (−0.0198 − 0.999i)10-s + (0.996 + 0.0794i)11-s + (0.999 − 0.0397i)12-s + (0.980 − 0.197i)13-s + (−0.996 − 0.0794i)14-s + (0.921 + 0.387i)15-s + (−0.905 − 0.423i)16-s + (−0.293 − 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.348 - 0.937i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.348 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.760042031 - 1.223091942i\)
\(L(\frac12)\) \(\approx\) \(1.760042031 - 1.223091942i\)
\(L(1)\) \(\approx\) \(1.621497742 - 0.6236623406i\)
\(L(1)\) \(\approx\) \(1.621497742 - 0.6236623406i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (0.780 - 0.625i)T \)
3 \( 1 + (0.255 + 0.966i)T \)
5 \( 1 + (0.610 - 0.792i)T \)
7 \( 1 + (-0.727 - 0.685i)T \)
11 \( 1 + (0.996 + 0.0794i)T \)
13 \( 1 + (0.980 - 0.197i)T \)
17 \( 1 + (-0.293 - 0.955i)T \)
19 \( 1 + (-0.804 + 0.594i)T \)
23 \( 1 + (-0.545 + 0.838i)T \)
29 \( 1 + (0.727 - 0.685i)T \)
31 \( 1 + (0.700 - 0.714i)T \)
37 \( 1 + (0.578 + 0.815i)T \)
41 \( 1 + (-0.511 + 0.859i)T \)
43 \( 1 + (0.949 - 0.312i)T \)
47 \( 1 + (-0.700 + 0.714i)T \)
53 \( 1 + (0.804 - 0.594i)T \)
59 \( 1 + (0.949 + 0.312i)T \)
61 \( 1 + (0.641 + 0.767i)T \)
67 \( 1 + (-0.961 + 0.274i)T \)
71 \( 1 + (-0.921 + 0.387i)T \)
73 \( 1 + (0.888 - 0.459i)T \)
79 \( 1 + (0.216 + 0.976i)T \)
83 \( 1 + (-0.177 + 0.984i)T \)
89 \( 1 + (-0.780 + 0.625i)T \)
97 \( 1 + (-0.578 - 0.815i)T \)
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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

−25.31758065056487608991659776542, −24.573722227747378917871043677913, −23.55694597981618473661475145409, −22.76457318717109763545407871429, −21.983752928906404734384379496965, −21.24264887993936567720642482255, −19.8490514400178486550254521735, −19.01533506407452510715209561071, −18.013759658481239574063150045789, −17.33261353697004947890187412702, −16.18146288889031895739674744850, −15.03390700634691420693876490060, −14.34962408411178321678275297880, −13.51422901149072156385567569390, −12.73398265775592089292605723885, −11.88512101233268131356978430177, −10.747588606289441880240016870274, −9.01970379720560693616084870749, −8.41549612723215028940719470598, −6.81390048009080993596081745517, −6.474106429383797959571901611416, −5.773202318163029900684740700496, −3.92718743860722522095059482865, −2.89911779145257294884571944744, −1.93234933798322466429421166881, 1.13389406921456257904667479760, 2.63892351795150962676312875463, 3.89432538037044096526177110000, 4.40360701210414868972303245034, 5.70815257979787547255474775275, 6.487175686692970767925315363370, 8.431292787727030502246077931714, 9.61324797886043404673185685249, 9.92641960005902997726826838303, 11.132329162112212362921666705590, 12.07626012212532931949231025, 13.46195586635495239391028857099, 13.66946950316763273289130372364, 14.84574157728004397265034598202, 15.94238845021236675030574199855, 16.54533965643939714780116876080, 17.655998749089426034319414940949, 19.23532118726547545879302422324, 19.969439234040838817798534586062, 20.68891079919427961341964823876, 21.26431910293952253766041408772, 22.34220486644899625804351594225, 22.86091401647974805970471220334, 23.91587195850687926414864715740, 25.19495967482708571108160773830

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