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Properties

Label 1-1309-1309.1062-r0-0-0
Degree $1$
Conductor $1309$
Sign $0.564 - 0.825i$
Analytic cond. $6.07897$
Root an. cond. $6.07897$
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  + (−0.406 + 0.913i)2-s + (−0.544 − 0.838i)3-s + (−0.669 − 0.743i)4-s + (0.777 − 0.629i)5-s + (0.987 − 0.156i)6-s + (0.951 − 0.309i)8-s + (−0.406 + 0.913i)9-s + (0.258 + 0.965i)10-s + (−0.258 + 0.965i)12-s + (0.809 − 0.587i)13-s + (−0.951 − 0.309i)15-s + (−0.104 + 0.994i)16-s + (−0.669 − 0.743i)18-s + (−0.743 − 0.669i)19-s + (−0.987 − 0.156i)20-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.544 − 0.838i)3-s + (−0.669 − 0.743i)4-s + (0.777 − 0.629i)5-s + (0.987 − 0.156i)6-s + (0.951 − 0.309i)8-s + (−0.406 + 0.913i)9-s + (0.258 + 0.965i)10-s + (−0.258 + 0.965i)12-s + (0.809 − 0.587i)13-s + (−0.951 − 0.309i)15-s + (−0.104 + 0.994i)16-s + (−0.669 − 0.743i)18-s + (−0.743 − 0.669i)19-s + (−0.987 − 0.156i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(6.07897\)
Root analytic conductor: \(6.07897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1309} (1062, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1309,\ (0:\ ),\ 0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9657108183 - 0.5097049531i\)
\(L(\frac12)\) \(\approx\) \(0.9657108183 - 0.5097049531i\)
\(L(1)\) \(\approx\) \(0.8299301832 - 0.08120130655i\)
\(L(1)\) \(\approx\) \(0.8299301832 - 0.08120130655i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.406 + 0.913i)T \)
3 \( 1 + (-0.544 - 0.838i)T \)
5 \( 1 + (0.777 - 0.629i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 + (0.965 + 0.258i)T \)
29 \( 1 + (-0.891 + 0.453i)T \)
31 \( 1 + (0.629 - 0.777i)T \)
37 \( 1 + (0.838 + 0.544i)T \)
41 \( 1 + (0.891 + 0.453i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (-0.994 + 0.104i)T \)
59 \( 1 + (0.743 - 0.669i)T \)
61 \( 1 + (0.629 + 0.777i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.156 + 0.987i)T \)
73 \( 1 + (-0.0523 + 0.998i)T \)
79 \( 1 + (0.933 - 0.358i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.987 + 0.156i)T \)
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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

−21.06049518533883433112870667388, −20.74288271989779501352100561647, −19.48657305076085654411419840514, −18.756162042337937155432858830318, −18.06962565609522373024017770888, −17.35294919290684415663511360595, −16.747964554952612919232724406313, −15.97182680954656645442494467349, −14.81180637939882424505286306895, −14.19112496655010729190634350792, −13.23303176991068037575371181477, −12.45517852549519746981694990629, −11.42323935019450893338388561646, −10.86645388459547054763273904589, −10.378329208863431969372424277948, −9.38193092558906960441748927274, −9.055840006419509529068345276883, −7.86442367550637249931816316794, −6.61200886625769343263402894026, −5.903189879982894262996622765184, −4.82448629488795264028522493873, −3.97168471001213835309242888332, −3.16761140873515103523748411468, −2.188700296788375434659147477216, −1.08832739842073968279822797224, 0.659124885456126911875032340294, 1.417385824864894840619796904400, 2.52177612386924667939316184881, 4.24106274056939062492089857666, 5.23021667975798104574679352357, 5.7717562013069675434776922592, 6.50409040384418029020168014475, 7.28096438652975497639623056476, 8.26574219557811431769316635733, 8.80668677979376585257087247108, 9.75623744764450723385013820872, 10.68502835372914363317976723383, 11.41299856282951222446065697262, 12.84501004335153735679610864178, 13.09367932032073917445928926179, 13.79560790181713090723471966184, 14.76752637156360188148127899035, 15.67243212885952565870650931036, 16.52423053207335322488547217492, 17.183724929003737288741691909685, 17.58366766637976918727883212879, 18.42000899248615639273905839810, 18.97650525561708424203255546219, 19.91827556637385476372961824005, 20.74755596363401573657837695885

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