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Properties

Label 1-317-317.149-r0-0-0
Degree $1$
Conductor $317$
Sign $0.277 - 0.960i$
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  + (−0.961 + 0.274i)2-s + (−0.980 + 0.197i)3-s + (0.848 − 0.528i)4-s + (−0.0992 − 0.995i)5-s + (0.888 − 0.459i)6-s + (0.216 − 0.976i)7-s + (−0.671 + 0.741i)8-s + (0.921 − 0.387i)9-s + (0.368 + 0.929i)10-s + (0.0596 + 0.998i)11-s + (−0.727 + 0.685i)12-s + (0.804 − 0.594i)13-s + (0.0596 + 0.998i)14-s + (0.293 + 0.955i)15-s + (0.441 − 0.897i)16-s + (0.578 − 0.815i)17-s + ⋯
L(s)  = 1  + (−0.961 + 0.274i)2-s + (−0.980 + 0.197i)3-s + (0.848 − 0.528i)4-s + (−0.0992 − 0.995i)5-s + (0.888 − 0.459i)6-s + (0.216 − 0.976i)7-s + (−0.671 + 0.741i)8-s + (0.921 − 0.387i)9-s + (0.368 + 0.929i)10-s + (0.0596 + 0.998i)11-s + (−0.727 + 0.685i)12-s + (0.804 − 0.594i)13-s + (0.0596 + 0.998i)14-s + (0.293 + 0.955i)15-s + (0.441 − 0.897i)16-s + (0.578 − 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.277 - 0.960i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.277 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4617292317 - 0.3471294051i\)
\(L(\frac12)\) \(\approx\) \(0.4617292317 - 0.3471294051i\)
\(L(1)\) \(\approx\) \(0.5571601249 - 0.1198183937i\)
\(L(1)\) \(\approx\) \(0.5571601249 - 0.1198183937i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (-0.961 + 0.274i)T \)
3 \( 1 + (-0.980 + 0.197i)T \)
5 \( 1 + (-0.0992 - 0.995i)T \)
7 \( 1 + (0.216 - 0.976i)T \)
11 \( 1 + (0.0596 + 0.998i)T \)
13 \( 1 + (0.804 - 0.594i)T \)
17 \( 1 + (0.578 - 0.815i)T \)
19 \( 1 + (0.888 + 0.459i)T \)
23 \( 1 + (-0.999 + 0.0397i)T \)
29 \( 1 + (0.216 + 0.976i)T \)
31 \( 1 + (-0.827 - 0.561i)T \)
37 \( 1 + (0.754 - 0.656i)T \)
41 \( 1 + (0.700 - 0.714i)T \)
43 \( 1 + (0.971 + 0.236i)T \)
47 \( 1 + (-0.827 - 0.561i)T \)
53 \( 1 + (0.888 + 0.459i)T \)
59 \( 1 + (0.971 - 0.236i)T \)
61 \( 1 + (-0.610 - 0.792i)T \)
67 \( 1 + (-0.545 - 0.838i)T \)
71 \( 1 + (0.293 - 0.955i)T \)
73 \( 1 + (-0.936 - 0.350i)T \)
79 \( 1 + (0.848 + 0.528i)T \)
83 \( 1 + (-0.255 + 0.966i)T \)
89 \( 1 + (-0.961 + 0.274i)T \)
97 \( 1 + (0.754 - 0.656i)T \)
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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

−25.55746232351470919368618121050, −24.44107805848245328789991363837, −23.75651575554743771898857873445, −22.45629834957666474847530586547, −21.68151150795548669566175209894, −21.19723729956304968007594618223, −19.51287959401196068662804229425, −18.80777549293220310990351489310, −18.24397989879748817660478675008, −17.57254564351773689056951199782, −16.28080244136575618334978204930, −15.84385571657453310747987997053, −14.6075044381139208673856241231, −13.218157351312670355374036704919, −11.8571426834679085556210086168, −11.497471233148233349991022655860, −10.66321590324178568729180892946, −9.68042823826839878683141538917, −8.44402251167266113347346874838, −7.49587922371528000611128217935, −6.25958083416316730475182622085, −5.84912079084733156449889629338, −3.837909568861251498507508255356, −2.584202176070279963412859677072, −1.30823570542765243731424167287, 0.67578091909481641707748551475, 1.59204082464835166079676833509, 3.852381385071347105999304479474, 5.0635266067735435109497854857, 5.89823336546897098513807698354, 7.25604415134915274785254694666, 7.84583816858840426528755134000, 9.31043493375010893834214488038, 10.010724094592085631003093405414, 10.934989503587257935807907502515, 11.89421168388480576926564512781, 12.7503206622467116331764753035, 14.14389462931853090327940380410, 15.48888598353189612402385879097, 16.34080541437350041259002745484, 16.69939296976096509922510933031, 17.97337820543746714488705453604, 18.04088548240174226017878362543, 19.76328315548238804019679848199, 20.47531804234960769696184920876, 20.9961115718899032059432539817, 22.644816523770689408369085430313, 23.39683782433993671510695206726, 24.07813367322892818489327109560, 24.99522611974456146216356116161

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