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Properties

Label 1-349-349.123-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.417 + 0.908i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.417 + 0.908i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.417 + 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4929277971 - 0.7688550983i\)
\(L(\frac12)\) \(\approx\) \(-0.4929277971 - 0.7688550983i\)
\(L(1)\) \(\approx\) \(0.3671835822 - 0.8471390037i\)
\(L(1)\) \(\approx\) \(0.3671835822 - 0.8471390037i\)

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

−25.7249635421033454346581244646, −24.36694539541166056431624472127, −23.41996148560991646088444257849, −22.99421158294907157792853019081, −21.97917673494752116432096415977, −21.33796585314168214092759345792, −20.67439103934486681429107052978, −18.68483289173941023107927728767, −18.38817714308919806432326257559, −17.2128871185032745675871603142, −16.09211078341204778761700141948, −15.75572951914795568039329873515, −14.53117370569656131941829173473, −14.43387508827277012367820058071, −12.64892100045761899890963053429, −11.82594619439009860667404544018, −10.95618691003986655861451832913, −9.83746888562302178292366831916, −8.57021986813284558218830240487, −7.79927010005306386641670341354, −6.430564678656384023428155270902, −5.73109585499552207353072870076, −4.68051190779840762814954215553, −3.73352732158416016923830489734, −2.644518486575431531987449096853, 0.541473433799375211848029210504, 1.44206365064008502631879923378, 2.87642761792729003938352001232, 4.260836254397137739009387025656, 5.15463222140262957441045379333, 6.01399218063107017337504352416, 7.680978636983753032462144803414, 8.17746318811565873924640335075, 9.76985006004333349261990270205, 10.88573100897445129036973412351, 11.46464714175976103703459428616, 12.53596949762418871871610753020, 13.17279341248971210616497322443, 13.781179306476460810636453282816, 15.13294980899901668140349125068, 16.218495916339458567511099064415, 17.40899806433093357718900144531, 18.03178553057029652518576471623, 19.174486459697121709122909656, 19.82507227702656477812968808853, 20.71061160330684695872460288233, 21.36156920385971732895503470871, 22.89479890805768351150348872691, 23.27073425170370768105343461976, 23.95266194362979901011272736019

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