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Properties

Label 1-349-349.109-r0-0-0
Degree $1$
Conductor $349$
Sign $0.829 - 0.558i$
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  + (−0.997 + 0.0721i)2-s + (−0.302 − 0.953i)3-s + (0.989 − 0.143i)4-s + (−0.773 + 0.633i)5-s + (0.370 + 0.928i)6-s + (−0.126 − 0.992i)7-s + (−0.976 + 0.214i)8-s + (−0.817 + 0.576i)9-s + (0.725 − 0.687i)10-s + (−0.0541 + 0.998i)11-s + (−0.436 − 0.899i)12-s + (−0.750 + 0.661i)13-s + (0.197 + 0.980i)14-s + (0.837 + 0.546i)15-s + (0.958 − 0.284i)16-s + (0.976 + 0.214i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0721i)2-s + (−0.302 − 0.953i)3-s + (0.989 − 0.143i)4-s + (−0.773 + 0.633i)5-s + (0.370 + 0.928i)6-s + (−0.126 − 0.992i)7-s + (−0.976 + 0.214i)8-s + (−0.817 + 0.576i)9-s + (0.725 − 0.687i)10-s + (−0.0541 + 0.998i)11-s + (−0.436 − 0.899i)12-s + (−0.750 + 0.661i)13-s + (0.197 + 0.980i)14-s + (0.837 + 0.546i)15-s + (0.958 − 0.284i)16-s + (0.976 + 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.829 - 0.558i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.829 - 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5286659317 - 0.1612927597i\)
\(L(\frac12)\) \(\approx\) \(0.5286659317 - 0.1612927597i\)
\(L(1)\) \(\approx\) \(0.5417309812 - 0.1060169790i\)
\(L(1)\) \(\approx\) \(0.5417309812 - 0.1060169790i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.997 + 0.0721i)T \)
3 \( 1 + (-0.302 - 0.953i)T \)
5 \( 1 + (-0.773 + 0.633i)T \)
7 \( 1 + (-0.126 - 0.992i)T \)
11 \( 1 + (-0.0541 + 0.998i)T \)
13 \( 1 + (-0.750 + 0.661i)T \)
17 \( 1 + (0.976 + 0.214i)T \)
19 \( 1 + (0.530 - 0.847i)T \)
23 \( 1 + (0.750 - 0.661i)T \)
29 \( 1 + (-0.436 + 0.899i)T \)
31 \( 1 + (0.267 + 0.963i)T \)
37 \( 1 + (-0.947 - 0.319i)T \)
41 \( 1 + (0.976 - 0.214i)T \)
43 \( 1 + (0.891 - 0.452i)T \)
47 \( 1 + (-0.647 - 0.762i)T \)
53 \( 1 + (0.994 + 0.108i)T \)
59 \( 1 + (0.302 + 0.953i)T \)
61 \( 1 + (0.561 - 0.827i)T \)
67 \( 1 + (0.468 - 0.883i)T \)
71 \( 1 + (0.302 - 0.953i)T \)
73 \( 1 + (-0.922 - 0.386i)T \)
79 \( 1 + (0.994 - 0.108i)T \)
83 \( 1 + (0.935 + 0.353i)T \)
89 \( 1 + (0.0180 - 0.999i)T \)
97 \( 1 + (0.773 + 0.633i)T \)
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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

−24.90204282698542386894263413058, −24.39493173447383620444983909569, −23.091199722160568823204590934403, −22.15069842351224837854354304233, −21.02834741028839533202557628475, −20.69852186329179235021962781271, −19.378058321757629684529419843721, −18.95061240624730487374992032031, −17.65197275631843283538799781745, −16.750284998535574867757504699705, −16.10540809961937740247738582457, −15.4287989992220276780684786834, −14.62009141674001016164152636424, −12.69833675561692084969089646288, −11.78152997181404820634727847736, −11.302177624473014988724298728563, −9.99895163163286458475853260421, −9.32372898916540479796204363285, −8.385643676265636316887554552727, −7.64262136911923911984676529573, −5.89086578085675333612967498693, −5.3071311676244544637105250867, −3.653382602640497933101761613527, −2.788404379089699622666591780885, −0.80230777062087605984292940563, 0.770364225390284021757013250019, 2.120974687858435233769809481, 3.31399270260413341076476729798, 4.98085316638590583023153514775, 6.63502330121695770605326785757, 7.21827202088446239694696368321, 7.60719202652131135361926255140, 8.93725475433408030296839923284, 10.23681995223254343083024609507, 10.924758799017646338055121645470, 11.93709617965594465713241035301, 12.5932199278700900730814515715, 14.13531024277316674555540418802, 14.85066466615412700108286498637, 16.12825086913950097773343933586, 16.93561182050007201404814702448, 17.7203172090429160923132582032, 18.51778452210810364357480054641, 19.5094276067350134807679645285, 19.74047887299677702281313331581, 20.88713375513956488092686540712, 22.425515032118800143790895104559, 23.26366919798806521559760257158, 23.895931319264082723071567564424, 24.75226586660521283723430648628

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