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Properties

Label 1-1309-1309.395-r0-0-0
Degree $1$
Conductor $1309$
Sign $0.710 - 0.704i$
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  + (−0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s + i·6-s + 8-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)10-s + (−0.866 − 0.5i)12-s + 13-s − 15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s + i·20-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s + i·6-s + 8-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)10-s + (−0.866 − 0.5i)12-s + 13-s − 15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s + i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $0.710 - 0.704i$
Analytic conductor: \(6.07897\)
Root analytic conductor: \(6.07897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1309} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1309,\ (0:\ ),\ 0.710 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.245997484 - 0.5130016881i\)
\(L(\frac12)\) \(\approx\) \(1.245997484 - 0.5130016881i\)
\(L(1)\) \(\approx\) \(0.9949595285 - 0.03918033862i\)
\(L(1)\) \(\approx\) \(0.9949595285 - 0.03918033862i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + iT \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + iT \)
43 \( 1 + T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 - iT \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + iT \)
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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

−20.87983017670895400756043344744, −20.280770159529092319408478395840, −19.52752839737884019395594744987, −18.98918512083174150489343791598, −18.346773004402377509832325638945, −17.47839920483180233038305485121, −16.10503066338755438588210980231, −16.02553366457259963339756561615, −14.91244849215244952352488666504, −13.9998234851530708067174745784, −13.47578650575160234918731953110, −12.35497639083560095312493612684, −11.651443447353071193693246091352, −10.796037916194037454177916895377, −10.22051211025839405130258677667, −9.37287066947878562633153814548, −8.511638296449414678862991886349, −7.903912872647843152834352303397, −7.28733817688936766895692472375, −5.849837203554390057874341595891, −4.34079105723743800052142221926, −3.92383435333345850129932770364, −3.13285061051091237828051303438, −2.320439034634394733797898262018, −1.148897214307540121707805157539, 0.680040230215921529737736224798, 1.52793849744011152077151398352, 2.91820007256507304987487129001, 4.011547321723262371543463648515, 4.72306692267163076534228448089, 5.98411734824089349420358590857, 6.76708589226365395758812725889, 7.65507152360728498269819029999, 8.16567022155980972588310665099, 8.88364211672084488904525893831, 9.480099164365555600716442779663, 10.61763792625873172725517526317, 11.582959108175345441245328094791, 12.56976965113252878624871868995, 13.41389376477924514422881807293, 13.98329844865826860936475493663, 14.95201830865988972473598630565, 15.54703232474263706834198744642, 16.13461658379959747363627149874, 16.972781795590992326075208547693, 18.079030985921333683252261159, 18.47361718823038546191794762535, 19.284160422958225555732124605140, 20.0905770676599834572411781831, 20.34157752584007048698321145532

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