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Properties

Label 1-349-349.26-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.145 - 0.989i$
Analytic cond. $1.62074$
Root an. cond. $1.62074$
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  + (0.197 − 0.980i)2-s + (−0.436 + 0.899i)3-s + (−0.922 − 0.386i)4-s + (−0.891 + 0.452i)5-s + (0.796 + 0.605i)6-s + (−0.674 + 0.738i)7-s + (−0.561 + 0.827i)8-s + (−0.619 − 0.785i)9-s + (0.267 + 0.963i)10-s + (−0.856 + 0.515i)11-s + (0.750 − 0.661i)12-s + (0.403 − 0.915i)13-s + (0.590 + 0.806i)14-s + (−0.0180 − 0.999i)15-s + (0.700 + 0.713i)16-s + (−0.561 − 0.827i)17-s + ⋯
L(s)  = 1  + (0.197 − 0.980i)2-s + (−0.436 + 0.899i)3-s + (−0.922 − 0.386i)4-s + (−0.891 + 0.452i)5-s + (0.796 + 0.605i)6-s + (−0.674 + 0.738i)7-s + (−0.561 + 0.827i)8-s + (−0.619 − 0.785i)9-s + (0.267 + 0.963i)10-s + (−0.856 + 0.515i)11-s + (0.750 − 0.661i)12-s + (0.403 − 0.915i)13-s + (0.590 + 0.806i)14-s + (−0.0180 − 0.999i)15-s + (0.700 + 0.713i)16-s + (−0.561 − 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.145 - 0.989i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.145 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3228097511 - 0.3736639876i\)
\(L(\frac12)\) \(\approx\) \(0.3228097511 - 0.3736639876i\)
\(L(1)\) \(\approx\) \(0.6126113849 - 0.1561104261i\)
\(L(1)\) \(\approx\) \(0.6126113849 - 0.1561104261i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.197 - 0.980i)T \)
3 \( 1 + (-0.436 + 0.899i)T \)
5 \( 1 + (-0.891 + 0.452i)T \)
7 \( 1 + (-0.674 + 0.738i)T \)
11 \( 1 + (-0.856 + 0.515i)T \)
13 \( 1 + (0.403 - 0.915i)T \)
17 \( 1 + (-0.561 - 0.827i)T \)
19 \( 1 + (0.935 - 0.353i)T \)
23 \( 1 + (0.403 - 0.915i)T \)
29 \( 1 + (0.750 + 0.661i)T \)
31 \( 1 + (0.907 - 0.419i)T \)
37 \( 1 + (-0.994 + 0.108i)T \)
41 \( 1 + (-0.561 + 0.827i)T \)
43 \( 1 + (0.874 + 0.484i)T \)
47 \( 1 + (-0.725 - 0.687i)T \)
53 \( 1 + (0.468 - 0.883i)T \)
59 \( 1 + (-0.436 + 0.899i)T \)
61 \( 1 + (-0.947 - 0.319i)T \)
67 \( 1 + (-0.161 - 0.986i)T \)
71 \( 1 + (-0.436 - 0.899i)T \)
73 \( 1 + (-0.302 - 0.953i)T \)
79 \( 1 + (0.468 + 0.883i)T \)
83 \( 1 + (0.837 + 0.546i)T \)
89 \( 1 + (0.336 - 0.941i)T \)
97 \( 1 + (-0.891 - 0.452i)T \)
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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

−24.74756933850855177963443544503, −24.10496476126297934984976281180, −23.36782116562782711952741526805, −23.0469426573600293537024664178, −21.90565929327063671579614650479, −20.66607200487210021773573485489, −19.219385090792870413427606891683, −19.063039524503318695548433249838, −17.72134589818589981079675763117, −16.966154582512206348891311495905, −16.07961620192899627939309217005, −15.6124837905241116045568337220, −13.95532504336434487623704874795, −13.47720087455549075519953722273, −12.5866536232770262269929156080, −11.69978725975335277876969075461, −10.49517555651713404074559678053, −8.97494188408655200548166418881, −8.06315888677998758498627623696, −7.31185090368454505559337367166, −6.48182655036159644105994424661, −5.45654417229301240193920223791, −4.312960018729130440774362509527, −3.23412125868318747254059009051, −1.04962636098400456459828968380, 0.380790575404171447905836392238, 2.82341248299599120103741950431, 3.17397936944648507154742386819, 4.57561035881083848710434067272, 5.27082257238581632228950488557, 6.56687983378865800019553644177, 8.19108629062272444873601969531, 9.19139089049942968528745858756, 10.1780581485851673243331980332, 10.8303909176599887622683532385, 11.80201763592590301065124452013, 12.43880404429119693448559446293, 13.57939825138473032263465509680, 14.97327368075250296590091805013, 15.48416507736156303765548817090, 16.28174244772835126949857682256, 17.97371258265529066790643788706, 18.26599924552298411583002378744, 19.51964763004062120911648356122, 20.288130871481527421230779177609, 21.02418778091550730068529240778, 22.11153643293329979619576922303, 22.83326208565131094235902573351, 22.97305261828668557019995255238, 24.35762338955580019544333575531

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