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Properties

Label 1-349-349.151-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.710 - 0.703i$
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  + (−0.232 − 0.972i)2-s + (−0.983 − 0.179i)3-s + (−0.891 + 0.452i)4-s + (0.126 − 0.992i)5-s + (0.0541 + 0.998i)6-s + (−0.0180 − 0.999i)7-s + (0.647 + 0.762i)8-s + (0.935 + 0.353i)9-s + (−0.994 + 0.108i)10-s + (0.976 − 0.214i)11-s + (0.958 − 0.284i)12-s + (0.700 + 0.713i)13-s + (−0.968 + 0.250i)14-s + (−0.302 + 0.953i)15-s + (0.590 − 0.806i)16-s + (0.647 − 0.762i)17-s + ⋯
L(s)  = 1  + (−0.232 − 0.972i)2-s + (−0.983 − 0.179i)3-s + (−0.891 + 0.452i)4-s + (0.126 − 0.992i)5-s + (0.0541 + 0.998i)6-s + (−0.0180 − 0.999i)7-s + (0.647 + 0.762i)8-s + (0.935 + 0.353i)9-s + (−0.994 + 0.108i)10-s + (0.976 − 0.214i)11-s + (0.958 − 0.284i)12-s + (0.700 + 0.713i)13-s + (−0.968 + 0.250i)14-s + (−0.302 + 0.953i)15-s + (0.590 − 0.806i)16-s + (0.647 − 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.710 - 0.703i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.710 - 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3417411475 - 0.8304703145i\)
\(L(\frac12)\) \(\approx\) \(0.3417411475 - 0.8304703145i\)
\(L(1)\) \(\approx\) \(0.5752790083 - 0.5456503844i\)
\(L(1)\) \(\approx\) \(0.5752790083 - 0.5456503844i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.232 - 0.972i)T \)
3 \( 1 + (-0.983 - 0.179i)T \)
5 \( 1 + (0.126 - 0.992i)T \)
7 \( 1 + (-0.0180 - 0.999i)T \)
11 \( 1 + (0.976 - 0.214i)T \)
13 \( 1 + (0.700 + 0.713i)T \)
17 \( 1 + (0.647 - 0.762i)T \)
19 \( 1 + (0.989 - 0.143i)T \)
23 \( 1 + (0.700 + 0.713i)T \)
29 \( 1 + (0.958 + 0.284i)T \)
31 \( 1 + (0.468 + 0.883i)T \)
37 \( 1 + (0.267 - 0.963i)T \)
41 \( 1 + (0.647 + 0.762i)T \)
43 \( 1 + (-0.674 - 0.738i)T \)
47 \( 1 + (-0.947 + 0.319i)T \)
53 \( 1 + (0.907 - 0.419i)T \)
59 \( 1 + (-0.983 - 0.179i)T \)
61 \( 1 + (-0.725 - 0.687i)T \)
67 \( 1 + (-0.370 - 0.928i)T \)
71 \( 1 + (-0.983 + 0.179i)T \)
73 \( 1 + (0.874 + 0.484i)T \)
79 \( 1 + (0.907 + 0.419i)T \)
83 \( 1 + (-0.922 + 0.386i)T \)
89 \( 1 + (-0.436 + 0.899i)T \)
97 \( 1 + (0.126 + 0.992i)T \)
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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

−25.09294774832615140997975545158, −24.45250283780306817349906109257, −23.19708226775096146429774138728, −22.654695068325341883108104916912, −22.09980170466385816101517554229, −21.15789624446131860383980931976, −19.4043587180598185047632321875, −18.52797210229884019241088943814, −18.021495636899317064741168485699, −17.17535870098881540538568760781, −16.25221191837943699828832272976, −15.239552694198288320319899501474, −14.85174919467522191254964302300, −13.60115948143566889643315074552, −12.404910082886791690720439573293, −11.44758088375121360300156417158, −10.328506461560482272614676977930, −9.62221113716783031982957700214, −8.4016681538927592539322111963, −7.22346411214819215156979860070, −6.15716282601887227438128357779, −5.88287199084311021952685734739, −4.57352693770228983465442799337, −3.27109614855334901815829010286, −1.2765502864359383465388870419, 0.96862540239810526609927598558, 1.38977833930365731618977541391, 3.48573518688098679983729183457, 4.47655981946233439125395778137, 5.27879074621318580899938072286, 6.70860404368486507359278380565, 7.83558042802029574942344416025, 9.14257320250627525096682631228, 9.82658386628480880031725240231, 10.993830949414939355053951232864, 11.702468631300083566826144561248, 12.41664554204510863113563419961, 13.532005222957433219617878471202, 13.987186670657429471748219415311, 16.165930555846087259859711567616, 16.6150615658057898717741781126, 17.45038923777988663527619875026, 18.1812979710660462533745070737, 19.37300973547433825442018070, 20.02147167721530528840358940765, 21.10518654285374977408316846623, 21.59697214954862254367882905432, 22.93738771931588729344366983975, 23.25343370568357375317335032077, 24.3289971603356053126944227743

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