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Properties

Label 1-349-349.322-r0-0-0
Degree $1$
Conductor $349$
Sign $0.952 - 0.304i$
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  + (−0.161 − 0.986i)2-s + (0.468 − 0.883i)3-s + (−0.947 + 0.319i)4-s + (−0.725 + 0.687i)5-s + (−0.947 − 0.319i)6-s + (−0.994 − 0.108i)7-s + (0.468 + 0.883i)8-s + (−0.561 − 0.827i)9-s + (0.796 + 0.605i)10-s + (0.267 + 0.963i)11-s + (−0.161 + 0.986i)12-s + (0.0541 + 0.998i)13-s + (0.0541 + 0.998i)14-s + (0.267 + 0.963i)15-s + (0.796 − 0.605i)16-s + (0.468 − 0.883i)17-s + ⋯
L(s)  = 1  + (−0.161 − 0.986i)2-s + (0.468 − 0.883i)3-s + (−0.947 + 0.319i)4-s + (−0.725 + 0.687i)5-s + (−0.947 − 0.319i)6-s + (−0.994 − 0.108i)7-s + (0.468 + 0.883i)8-s + (−0.561 − 0.827i)9-s + (0.796 + 0.605i)10-s + (0.267 + 0.963i)11-s + (−0.161 + 0.986i)12-s + (0.0541 + 0.998i)13-s + (0.0541 + 0.998i)14-s + (0.267 + 0.963i)15-s + (0.796 − 0.605i)16-s + (0.468 − 0.883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.952 - 0.304i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (322, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.952 - 0.304i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8014476426 - 0.1248249710i\)
\(L(\frac12)\) \(\approx\) \(0.8014476426 - 0.1248249710i\)
\(L(1)\) \(\approx\) \(0.7411144846 - 0.3254589745i\)
\(L(1)\) \(\approx\) \(0.7411144846 - 0.3254589745i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.161 - 0.986i)T \)
3 \( 1 + (0.468 - 0.883i)T \)
5 \( 1 + (-0.725 + 0.687i)T \)
7 \( 1 + (-0.994 - 0.108i)T \)
11 \( 1 + (0.267 + 0.963i)T \)
13 \( 1 + (0.0541 + 0.998i)T \)
17 \( 1 + (0.468 - 0.883i)T \)
19 \( 1 + (0.647 + 0.762i)T \)
23 \( 1 + (0.0541 + 0.998i)T \)
29 \( 1 + (-0.161 - 0.986i)T \)
31 \( 1 + (0.976 - 0.214i)T \)
37 \( 1 + (0.0541 + 0.998i)T \)
41 \( 1 + (0.468 + 0.883i)T \)
43 \( 1 + (0.267 + 0.963i)T \)
47 \( 1 + (-0.370 + 0.928i)T \)
53 \( 1 + (-0.856 + 0.515i)T \)
59 \( 1 + (0.468 - 0.883i)T \)
61 \( 1 + (-0.161 + 0.986i)T \)
67 \( 1 + (0.647 - 0.762i)T \)
71 \( 1 + (0.468 + 0.883i)T \)
73 \( 1 + (-0.994 - 0.108i)T \)
79 \( 1 + (-0.856 - 0.515i)T \)
83 \( 1 + (-0.725 + 0.687i)T \)
89 \( 1 + (0.907 + 0.419i)T \)
97 \( 1 + (-0.725 - 0.687i)T \)
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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

−24.91385558778848051709367596368, −24.227898834438799949422138947735, −23.094503418354871200966315476727, −22.39705592263532140358154494252, −21.57938069048801150096977559656, −20.28113618458087373556062868220, −19.51406357882056674364775842672, −18.86989646358158031066452188604, −17.36316481741687574414546006203, −16.47646891361090317585176575612, −15.9623262897147201861344015339, −15.318722253730494223368083082, −14.32467993867240099672589383516, −13.29192678595845353625373751805, −12.451494457878136981812598952809, −10.85035156812198519604743855111, −9.93299948356324951503768246669, −8.82182794244452121413290999590, −8.47092596234262397226940988859, −7.309937531414443724317529671975, −5.92381061326886347118555937001, −5.1072071414901273443156201552, −3.89962844296252157976804877023, −3.189450195182789321987272884760, −0.58071354909270008603178731719, 1.25625450714902940043494498882, 2.59574435082729087297443982169, 3.35231467683452320817441689785, 4.36758700337409063178436978321, 6.23967848145612000299974275683, 7.28986795934280619358172602297, 7.9864269083632493304815732032, 9.461233918921578150396978564756, 9.8398543047713193199947422498, 11.52635315062859768919595154809, 11.88646887283763453290222078823, 12.86385756305164479321231531087, 13.83182804827559779700005411744, 14.52010453725156003217758146450, 15.78979580452278320362415891366, 17.0630199704152221120025555665, 18.12688432982229823158984252524, 18.889068257185973963017124896100, 19.3564778579611185282402021899, 20.140671125690030574577768402386, 20.98913289109088090083412099194, 22.37596759270073463554173802655, 22.96829421779415517469603042826, 23.52145676013365526235268104989, 24.95035196144022004532690374071

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