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Properties

Label 1-349-349.118-r0-0-0
Degree $1$
Conductor $349$
Sign $0.405 + 0.914i$
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  + (0.468 − 0.883i)2-s + (−0.994 − 0.108i)3-s + (−0.561 − 0.827i)4-s + (0.647 − 0.762i)5-s + (−0.561 + 0.827i)6-s + (−0.947 + 0.319i)7-s + (−0.994 + 0.108i)8-s + (0.976 + 0.214i)9-s + (−0.370 − 0.928i)10-s + (−0.725 + 0.687i)11-s + (0.468 + 0.883i)12-s + (−0.161 + 0.986i)13-s + (−0.161 + 0.986i)14-s + (−0.725 + 0.687i)15-s + (−0.370 + 0.928i)16-s + (−0.994 − 0.108i)17-s + ⋯
L(s)  = 1  + (0.468 − 0.883i)2-s + (−0.994 − 0.108i)3-s + (−0.561 − 0.827i)4-s + (0.647 − 0.762i)5-s + (−0.561 + 0.827i)6-s + (−0.947 + 0.319i)7-s + (−0.994 + 0.108i)8-s + (0.976 + 0.214i)9-s + (−0.370 − 0.928i)10-s + (−0.725 + 0.687i)11-s + (0.468 + 0.883i)12-s + (−0.161 + 0.986i)13-s + (−0.161 + 0.986i)14-s + (−0.725 + 0.687i)15-s + (−0.370 + 0.928i)16-s + (−0.994 − 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.405 + 0.914i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.405 + 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1793755647 + 0.1166824551i\)
\(L(\frac12)\) \(\approx\) \(0.1793755647 + 0.1166824551i\)
\(L(1)\) \(\approx\) \(0.5855203249 - 0.3083003149i\)
\(L(1)\) \(\approx\) \(0.5855203249 - 0.3083003149i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.468 - 0.883i)T \)
3 \( 1 + (-0.994 - 0.108i)T \)
5 \( 1 + (0.647 - 0.762i)T \)
7 \( 1 + (-0.947 + 0.319i)T \)
11 \( 1 + (-0.725 + 0.687i)T \)
13 \( 1 + (-0.161 + 0.986i)T \)
17 \( 1 + (-0.994 - 0.108i)T \)
19 \( 1 + (-0.856 - 0.515i)T \)
23 \( 1 + (-0.161 + 0.986i)T \)
29 \( 1 + (0.468 - 0.883i)T \)
31 \( 1 + (0.796 + 0.605i)T \)
37 \( 1 + (-0.161 + 0.986i)T \)
41 \( 1 + (-0.994 + 0.108i)T \)
43 \( 1 + (-0.725 + 0.687i)T \)
47 \( 1 + (0.907 + 0.419i)T \)
53 \( 1 + (0.0541 - 0.998i)T \)
59 \( 1 + (-0.994 - 0.108i)T \)
61 \( 1 + (0.468 + 0.883i)T \)
67 \( 1 + (-0.856 + 0.515i)T \)
71 \( 1 + (-0.994 + 0.108i)T \)
73 \( 1 + (-0.947 + 0.319i)T \)
79 \( 1 + (0.0541 + 0.998i)T \)
83 \( 1 + (0.647 - 0.762i)T \)
89 \( 1 + (0.267 - 0.963i)T \)
97 \( 1 + (0.647 + 0.762i)T \)
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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

−24.66806712497333690507029135462, −23.555646680317321638386133801598, −22.9675950908825196796000398550, −22.14081884692638272479383601673, −21.758552159649218067464549992869, −20.582174938053649872035251622629, −18.91897162047106933951892148389, −18.21375203438835316193515893125, −17.3549497017055404979387688112, −16.63666054558195922252834410627, −15.72983689844604064444089512104, −15.02432238911346254191844545912, −13.69910048462748721730969343097, −13.06659318022399533385217743852, −12.23764459632278844722481321771, −10.66921582941276819972410709405, −10.28340112052793722983586946683, −8.8828177455126380400432993659, −7.523416043385351480557989251145, −6.48715674664193104638686274685, −6.0786463590125513885615671758, −5.061155166257115416866383389848, −3.786395146350650706426444576270, −2.657239829290350641904374223969, −0.1229821309870445676522828639, 1.56917849605321512718298901226, 2.55343607117588982703094524975, 4.33435893459463647776169693421, 4.93252933317870622774605028791, 6.036242505911384448424370570175, 6.770512110576731716895044855792, 8.74464964325559829210283465792, 9.74188983602574924670009185736, 10.27844310900339902038770456050, 11.58493035090128327619190695972, 12.24059968096094511570231201180, 13.21490666054119684360289938603, 13.49231181794925408950065648219, 15.26456263599191358005712129106, 16.00336343295689788870330327332, 17.21461428492022872374113581086, 17.86728413661756605449704054758, 18.912839928915396852902453533820, 19.69403467364863864106017151869, 20.839235610354637037416622174759, 21.62547575985140796527868751901, 22.14206831405722968175493991312, 23.23291352440523724068324233507, 23.772089980627314456437707009124, 24.72006385002793295203939966777

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