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Properties

Label 1-1309-1309.41-r1-0-0
Degree $1$
Conductor $1309$
Sign $0.999 + 0.00154i$
Analytic cond. $140.671$
Root an. cond. $140.671$
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  + (0.891 − 0.453i)2-s + (−0.852 − 0.522i)3-s + (0.587 − 0.809i)4-s + (0.649 + 0.760i)5-s + (−0.996 − 0.0784i)6-s + (0.156 − 0.987i)8-s + (0.453 + 0.891i)9-s + (0.923 + 0.382i)10-s + (−0.923 + 0.382i)12-s + (−0.951 − 0.309i)13-s + (−0.156 − 0.987i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)18-s + (0.987 + 0.156i)19-s + (0.996 − 0.0784i)20-s + ⋯
L(s)  = 1  + (0.891 − 0.453i)2-s + (−0.852 − 0.522i)3-s + (0.587 − 0.809i)4-s + (0.649 + 0.760i)5-s + (−0.996 − 0.0784i)6-s + (0.156 − 0.987i)8-s + (0.453 + 0.891i)9-s + (0.923 + 0.382i)10-s + (−0.923 + 0.382i)12-s + (−0.951 − 0.309i)13-s + (−0.156 − 0.987i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)18-s + (0.987 + 0.156i)19-s + (0.996 − 0.0784i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $0.999 + 0.00154i$
Analytic conductor: \(140.671\)
Root analytic conductor: \(140.671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1309} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1309,\ (1:\ ),\ 0.999 + 0.00154i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.887841664 + 0.002234898438i\)
\(L(\frac12)\) \(\approx\) \(2.887841664 + 0.002234898438i\)
\(L(1)\) \(\approx\) \(1.460250376 - 0.3871896877i\)
\(L(1)\) \(\approx\) \(1.460250376 - 0.3871896877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.891 - 0.453i)T \)
3 \( 1 + (-0.852 - 0.522i)T \)
5 \( 1 + (0.649 + 0.760i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.987 + 0.156i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (0.972 + 0.233i)T \)
31 \( 1 + (-0.996 + 0.0784i)T \)
37 \( 1 + (-0.233 + 0.972i)T \)
41 \( 1 + (0.972 - 0.233i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (-0.891 + 0.453i)T \)
59 \( 1 + (0.987 - 0.156i)T \)
61 \( 1 + (-0.0784 + 0.996i)T \)
67 \( 1 - T \)
71 \( 1 + (0.760 - 0.649i)T \)
73 \( 1 + (0.972 + 0.233i)T \)
79 \( 1 + (0.760 + 0.649i)T \)
83 \( 1 + (-0.453 + 0.891i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.0784 + 0.996i)T \)
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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

−21.00006415149873258207381852255, −20.41476443027731305150547408340, −19.49421702910443281033167134142, −17.88966609681813229022999313885, −17.68138083398270493465561109948, −16.62644576615388328775608470992, −16.31756251860657953499557779680, −15.59996761873811892373648128212, −14.48882601569246697408164490647, −14.0622225877745029624703802619, −12.794407441838227995440759116648, −12.50165287558128333842507681901, −11.662457456829806840484166417945, −10.83407641197574149066329518453, −9.79752830696899823963524242508, −9.1516364847537526616842045089, −7.9890368140974542276136483650, −7.003658995531973700765800085078, −6.16576232324609259440222110496, −5.48033677431496918279342943790, −4.755528770198324544049972706078, −4.21989178943597706369579542322, −2.9698180748972625987419516868, −1.86172877638101876883791600281, −0.50714951989162204535692280700, 0.924003984680444950494707345732, 1.89449227115671508578339425567, 2.6882759882245077688882727477, 3.67483396383603906883294154148, 4.97896261967683512499105753376, 5.472796251799318741463955975488, 6.26964969922625811368822921366, 7.07218969657007705766987032009, 7.667498126954677570902335476503, 9.45721474316008406084457867359, 10.15252332159292411716843326450, 10.783113307554647244432001729635, 11.6293843369785840405920480137, 12.21317264812006526126562955469, 13.0418364531453547872520690977, 13.7984460476306370621417866500, 14.3498820862964172982054652409, 15.31297387918061892147696102986, 16.10054623570205977611409361493, 17.07226874678997769810257313935, 17.882298250607341792950698829013, 18.45066427116622450707772778848, 19.35020453198838120863915182151, 19.93024368196295048367174215717, 21.083943556352606891970219702364

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