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Properties

Label 1-317-317.204-r0-0-0
Degree $1$
Conductor $317$
Sign $0.151 + 0.988i$
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  + (−0.700 − 0.714i)2-s + (0.992 + 0.119i)3-s + (−0.0198 + 0.999i)4-s + (−0.0596 + 0.998i)5-s + (−0.610 − 0.792i)6-s + (−0.476 + 0.878i)7-s + (0.727 − 0.685i)8-s + (0.971 + 0.236i)9-s + (0.754 − 0.656i)10-s + (−0.961 + 0.274i)11-s + (−0.138 + 0.990i)12-s + (−0.641 + 0.767i)13-s + (0.961 − 0.274i)14-s + (−0.177 + 0.984i)15-s + (−0.999 − 0.0397i)16-s + (0.255 − 0.966i)17-s + ⋯
L(s)  = 1  + (−0.700 − 0.714i)2-s + (0.992 + 0.119i)3-s + (−0.0198 + 0.999i)4-s + (−0.0596 + 0.998i)5-s + (−0.610 − 0.792i)6-s + (−0.476 + 0.878i)7-s + (0.727 − 0.685i)8-s + (0.971 + 0.236i)9-s + (0.754 − 0.656i)10-s + (−0.961 + 0.274i)11-s + (−0.138 + 0.990i)12-s + (−0.641 + 0.767i)13-s + (0.961 − 0.274i)14-s + (−0.177 + 0.984i)15-s + (−0.999 − 0.0397i)16-s + (0.255 − 0.966i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.151 + 0.988i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.151 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7111262014 + 0.6101730034i\)
\(L(\frac12)\) \(\approx\) \(0.7111262014 + 0.6101730034i\)
\(L(1)\) \(\approx\) \(0.8719534470 + 0.1705738059i\)
\(L(1)\) \(\approx\) \(0.8719534470 + 0.1705738059i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (-0.700 - 0.714i)T \)
3 \( 1 + (0.992 + 0.119i)T \)
5 \( 1 + (-0.0596 + 0.998i)T \)
7 \( 1 + (-0.476 + 0.878i)T \)
11 \( 1 + (-0.961 + 0.274i)T \)
13 \( 1 + (-0.641 + 0.767i)T \)
17 \( 1 + (0.255 - 0.966i)T \)
19 \( 1 + (0.610 - 0.792i)T \)
23 \( 1 + (-0.331 + 0.943i)T \)
29 \( 1 + (0.476 + 0.878i)T \)
31 \( 1 + (-0.936 + 0.350i)T \)
37 \( 1 + (-0.980 + 0.197i)T \)
41 \( 1 + (-0.888 - 0.459i)T \)
43 \( 1 + (0.441 + 0.897i)T \)
47 \( 1 + (0.936 - 0.350i)T \)
53 \( 1 + (-0.610 + 0.792i)T \)
59 \( 1 + (0.441 - 0.897i)T \)
61 \( 1 + (0.996 + 0.0794i)T \)
67 \( 1 + (-0.827 + 0.561i)T \)
71 \( 1 + (0.177 + 0.984i)T \)
73 \( 1 + (-0.0992 + 0.995i)T \)
79 \( 1 + (-0.0198 - 0.999i)T \)
83 \( 1 + (0.987 + 0.158i)T \)
89 \( 1 + (0.700 + 0.714i)T \)
97 \( 1 + (0.980 - 0.197i)T \)
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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

−25.13871347283958146935247799328, −24.12526731750819249711049870512, −23.777637835624069499712425834310, −22.52261774059430046733368225128, −20.89351863355534689384342110492, −20.34028554288441859910061115221, −19.56454442597015193185365097118, −18.80061842512275815102651972072, −17.67100578609181134222262675240, −16.71164058578210325688729806129, −15.98919342239723675663137056330, −15.13350945599267281139471823745, −14.06549888866007567985239023068, −13.253504414584762013832524048385, −12.4017874802478819124721875990, −10.3988895644301664795385198897, −9.9993698213208189675623635235, −8.82483565063710946297127952551, −7.96627657407955404122139544274, −7.47790673680977003086848909652, −6.03604953634880297419941677022, −4.86536737220293034434104342316, −3.63412708153981701959517155930, −2.02570095216115597367755835667, −0.64943177670373403439044712484, 2.00662775918484407369896626787, 2.77002372953545813560759157691, 3.45894607724081499239617741252, 5.02500082350218325056674146272, 7.00571107729535665051869506581, 7.485642845938797463423258393746, 8.78335504854370807885747316869, 9.56800661496127263353228172178, 10.24739034978121387372996044290, 11.44235109270039624822297815958, 12.376850819143239410772637930775, 13.456934233117922489023013294155, 14.345199485613986692483744877614, 15.59794413264662525724347689128, 16.03890820165470201675093516728, 17.741167995776443638342380292063, 18.488086624185520572445772509936, 19.05026831365968642181193010666, 19.84599879282239930543722770291, 20.78887831660390545715256534042, 21.86164279394814436722382895830, 22.070198887101171774991738515499, 23.588375166495474923201702250574, 24.96524350024290068964844987135, 25.71000731649330163850737655276

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