Properties

Label 1-307-307.115-r0-0-0
Degree $1$
Conductor $307$
Sign $0.232 - 0.972i$
Analytic cond. $1.42570$
Root an. cond. $1.42570$
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  + (−0.273 − 0.961i)2-s + (−0.602 − 0.798i)3-s + (−0.850 + 0.526i)4-s + (−0.850 − 0.526i)5-s + (−0.602 + 0.798i)6-s + (0.0922 + 0.995i)7-s + (0.739 + 0.673i)8-s + (−0.273 + 0.961i)9-s + (−0.273 + 0.961i)10-s + (−0.602 + 0.798i)11-s + (0.932 + 0.361i)12-s + (0.445 − 0.895i)13-s + (0.932 − 0.361i)14-s + (0.0922 + 0.995i)15-s + (0.445 − 0.895i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.273 − 0.961i)2-s + (−0.602 − 0.798i)3-s + (−0.850 + 0.526i)4-s + (−0.850 − 0.526i)5-s + (−0.602 + 0.798i)6-s + (0.0922 + 0.995i)7-s + (0.739 + 0.673i)8-s + (−0.273 + 0.961i)9-s + (−0.273 + 0.961i)10-s + (−0.602 + 0.798i)11-s + (0.932 + 0.361i)12-s + (0.445 − 0.895i)13-s + (0.932 − 0.361i)14-s + (0.0922 + 0.995i)15-s + (0.445 − 0.895i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(307\)
Sign: $0.232 - 0.972i$
Analytic conductor: \(1.42570\)
Root analytic conductor: \(1.42570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{307} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 307,\ (0:\ ),\ 0.232 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4945134256 - 0.3901560193i\)
\(L(\frac12)\) \(\approx\) \(0.4945134256 - 0.3901560193i\)
\(L(1)\) \(\approx\) \(0.5350005024 - 0.3391269689i\)
\(L(1)\) \(\approx\) \(0.5350005024 - 0.3391269689i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad307 \( 1 \)
good2 \( 1 + (-0.273 - 0.961i)T \)
3 \( 1 + (-0.602 - 0.798i)T \)
5 \( 1 + (-0.850 - 0.526i)T \)
7 \( 1 + (0.0922 + 0.995i)T \)
11 \( 1 + (-0.602 + 0.798i)T \)
13 \( 1 + (0.445 - 0.895i)T \)
17 \( 1 + T \)
19 \( 1 + (0.445 - 0.895i)T \)
23 \( 1 + (-0.273 + 0.961i)T \)
29 \( 1 + (-0.273 - 0.961i)T \)
31 \( 1 + (-0.602 + 0.798i)T \)
37 \( 1 + (0.0922 + 0.995i)T \)
41 \( 1 + (0.0922 - 0.995i)T \)
43 \( 1 + (0.739 + 0.673i)T \)
47 \( 1 + (0.445 - 0.895i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.982 + 0.183i)T \)
61 \( 1 + (0.739 + 0.673i)T \)
67 \( 1 + (0.932 + 0.361i)T \)
71 \( 1 + (0.739 - 0.673i)T \)
73 \( 1 + (0.445 - 0.895i)T \)
79 \( 1 + (0.739 - 0.673i)T \)
83 \( 1 + (0.0922 - 0.995i)T \)
89 \( 1 + (0.932 + 0.361i)T \)
97 \( 1 + (0.445 + 0.895i)T \)
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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

−25.96014916689949475526824058897, −24.28154272552427770794877737176, −23.55657509088789939704372719493, −23.09338626618351338366267967817, −22.22971971982418401872087138794, −21.11876235938810697490393696160, −20.06270436857292982402940278623, −18.76117384994994072815507479043, −18.29246643158489733909762182241, −16.84977972820149804766683880075, −16.4017988605395359090265910994, −15.785210929846201452739946672564, −14.50574108808911981252845391436, −14.125476470757545950703044951188, −12.54587325819447719989043849786, −11.15205858670755928234806643670, −10.59743504749044103582052247317, −9.59588274023441942379317186803, −8.3272492811328074306902585538, −7.44380598145737735680923580228, −6.419867891968393163072975753546, −5.41750224948579568195208425513, −4.16955379306902155063062974022, −3.57322624238536124116034231418, −0.755916605622454197213550953969, 0.86718155066658259758193032355, 2.15491552467756690193148196596, 3.350929833053668813806296094964, 4.9154871762026363150161365484, 5.56541285418775487831658133307, 7.46797763224188232547492346608, 8.03771449133226815641789706398, 9.109762778860547976676155374536, 10.374315045569344028078428006094, 11.46272660991986587867402799253, 12.081873590995323179402139577016, 12.73394725132954766670325091731, 13.54622075125451455889347812479, 15.16294979126332349831302924443, 16.10429959599371940402012406957, 17.33191510788430554697088660038, 18.057142529914971719102657657386, 18.78549387118976680569026498705, 19.620392930611361981882129146751, 20.44336242623431055902156570934, 21.44413797742077094110508280750, 22.53220339783891834717485684978, 23.16633209650225630809885807453, 23.97964900627472433779781083098, 25.13478972346293409072151321286

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