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Properties

Label 1-349-349.122-r0-0-0
Degree $1$
Conductor $349$
Sign $0.998 - 0.0464i$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + 10-s + 11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + 10-s + 11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.998 - 0.0464i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.998 - 0.0464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6401520398 + 0.01488878742i\)
\(L(\frac12)\) \(\approx\) \(0.6401520398 + 0.01488878742i\)
\(L(1)\) \(\approx\) \(0.6247625002 - 0.03905588321i\)
\(L(1)\) \(\approx\) \(0.6247625002 - 0.03905588321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−24.90595792575171587717985963054, −24.04768810991613043789924592183, −23.45817144790547094848628719038, −22.51389235035939397463210041347, −21.58353644570482318682909902737, −19.868696348166199112778384617565, −19.259155469530153645322870244810, −18.742218054378696777306230000393, −17.44489242483316295240996613705, −16.91582471731416696213051596789, −16.12890792400107658218705540178, −15.175610493998912539525066171721, −14.05680548384739427921048327270, −13.075468870454995832897392498816, −12.056911437521085684741237117755, −11.463727134209272047540018420288, −9.723668658230368025662072517126, −8.98327036899254823579846057324, −8.04971689296618546098286024237, −7.09646291575420698992299669651, −6.15563331840131807954210201897, −5.343050163398876333205995770933, −4.18611058582543331990683722826, −2.07682550575883668428060584609, −0.80519720170064812969783163980, 0.8213689356517225892250104714, 2.84666859950018053455955900395, 3.71210524417728053144065984906, 4.36883418050281812994582798502, 6.0556550587542728837051186835, 7.22079798750949995213882827080, 8.27076787058420321273006179475, 9.66583546271957673420955924414, 10.23563505694534062080591238946, 10.88103898351647979303926417471, 11.89095969986767735630792301847, 12.6137382226248455597643090229, 14.17143966305332272975153168785, 14.838215524068980359675655075490, 16.26962849143045733321415378311, 16.81011769747296634388394924359, 17.711272794295047869388851108738, 18.7191193912611934886316726117, 19.70016258914340816296527022574, 20.26245581789064765038420861845, 21.30772102031617950273667029458, 22.30945297165517959173913906543, 22.71115070993083888738588837488, 23.4080136201969545781281348225, 25.1953666731524502218591361944

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