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Properties

Label 1-149-149.29-r0-0-0
Degree $1$
Conductor $149$
Sign $0.993 + 0.115i$
Analytic cond. $0.691953$
Root an. cond. $0.691953$
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  + (0.372 − 0.927i)2-s + (−0.967 − 0.251i)3-s + (−0.721 − 0.691i)4-s + (−0.450 + 0.892i)5-s + (−0.594 + 0.803i)6-s + (0.660 + 0.750i)7-s + (−0.911 + 0.411i)8-s + (0.873 + 0.487i)9-s + (0.660 + 0.750i)10-s + (−0.721 + 0.691i)11-s + (0.524 + 0.851i)12-s + (0.985 − 0.169i)13-s + (0.942 − 0.333i)14-s + (0.660 − 0.750i)15-s + (0.0424 + 0.999i)16-s + (−0.967 − 0.251i)17-s + ⋯
L(s)  = 1  + (0.372 − 0.927i)2-s + (−0.967 − 0.251i)3-s + (−0.721 − 0.691i)4-s + (−0.450 + 0.892i)5-s + (−0.594 + 0.803i)6-s + (0.660 + 0.750i)7-s + (−0.911 + 0.411i)8-s + (0.873 + 0.487i)9-s + (0.660 + 0.750i)10-s + (−0.721 + 0.691i)11-s + (0.524 + 0.851i)12-s + (0.985 − 0.169i)13-s + (0.942 − 0.333i)14-s + (0.660 − 0.750i)15-s + (0.0424 + 0.999i)16-s + (−0.967 − 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.993 + 0.115i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.993 + 0.115i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7832083750 + 0.04532660565i\)
\(L(\frac12)\) \(\approx\) \(0.7832083750 + 0.04532660565i\)
\(L(1)\) \(\approx\) \(0.8180217651 - 0.1639113392i\)
\(L(1)\) \(\approx\) \(0.8180217651 - 0.1639113392i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.372 - 0.927i)T \)
3 \( 1 + (-0.967 - 0.251i)T \)
5 \( 1 + (-0.450 + 0.892i)T \)
7 \( 1 + (0.660 + 0.750i)T \)
11 \( 1 + (-0.721 + 0.691i)T \)
13 \( 1 + (0.985 - 0.169i)T \)
17 \( 1 + (-0.967 - 0.251i)T \)
19 \( 1 + (0.778 + 0.628i)T \)
23 \( 1 + (0.985 + 0.169i)T \)
29 \( 1 + (-0.292 + 0.956i)T \)
31 \( 1 + (0.985 + 0.169i)T \)
37 \( 1 + (-0.721 + 0.691i)T \)
41 \( 1 + (0.0424 + 0.999i)T \)
43 \( 1 + (-0.828 + 0.559i)T \)
47 \( 1 + (0.778 - 0.628i)T \)
53 \( 1 + (-0.828 - 0.559i)T \)
59 \( 1 + (0.210 + 0.977i)T \)
61 \( 1 + (0.372 - 0.927i)T \)
67 \( 1 + (0.942 + 0.333i)T \)
71 \( 1 + (-0.450 + 0.892i)T \)
73 \( 1 + (-0.127 - 0.991i)T \)
79 \( 1 + (-0.127 + 0.991i)T \)
83 \( 1 + (-0.127 - 0.991i)T \)
89 \( 1 + (-0.996 + 0.0848i)T \)
97 \( 1 + (-0.828 - 0.559i)T \)
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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

−28.03411845692154407696923588802, −26.89077542686342092417071237089, −26.46894058101870836148805077275, −24.712554242618306624315122687908, −23.99267750430401018973543473205, −23.46961751269729312516761564179, −22.51889836290484815675451171358, −21.234077962820129344243412180, −20.65171660794380086817007531956, −18.81572079213723028672788585866, −17.59421541291729918990299193461, −17.01284211975645356723542040561, −15.91757595526761301414065951095, −15.49156669780762654377853156750, −13.70729618361030501185664361159, −13.04573814155314736057692574890, −11.70878844909284821258095238246, −10.81192857758172787409140849434, −9.095699170932761374462128526984, −8.05503434831298717454264664055, −6.899398592427760094211816421466, −5.59237600604975534693613626753, −4.72295197229134582269638367561, −3.81789472996379958549890470159, −0.7464069209382251149287256938, 1.607517062922305534717207915552, 2.99887092710881795354479876619, 4.5788735644679691292327596618, 5.54847951993244566837673046934, 6.82541071504795749143901660182, 8.31306549528364924927014551165, 9.97595745740370272965701181145, 11.009010479378171793458729893453, 11.534783634093737711909079671950, 12.56102128042703277160069920953, 13.65042404261054099879939465251, 15.02268227140115454470700552424, 15.77472178087257617836528524420, 17.72115214772090467884611598657, 18.28117581865029011581035193069, 18.906020420154850901144128024093, 20.369247435327301713015202255026, 21.3798966101899737505851741081, 22.32543326941636746774849385657, 23.02777905386470523825789003874, 23.753434386984102012036993010149, 24.95712284604391026759468628114, 26.59247797767421034463571858182, 27.50162692821013857341638602077, 28.30578806964099897295265006797

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