Properties

Label 1-149-149.37-r0-0-0
Degree $1$
Conductor $149$
Sign $0.580 + 0.814i$
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  + (−0.996 + 0.0848i)2-s + (−0.450 + 0.892i)3-s + (0.985 − 0.169i)4-s + (−0.828 − 0.559i)5-s + (0.372 − 0.927i)6-s + (0.873 + 0.487i)7-s + (−0.967 + 0.251i)8-s + (−0.594 − 0.803i)9-s + (0.873 + 0.487i)10-s + (0.985 + 0.169i)11-s + (−0.292 + 0.956i)12-s + (0.210 − 0.977i)13-s + (−0.911 − 0.411i)14-s + (0.873 − 0.487i)15-s + (0.942 − 0.333i)16-s + (−0.450 + 0.892i)17-s + ⋯
L(s)  = 1  + (−0.996 + 0.0848i)2-s + (−0.450 + 0.892i)3-s + (0.985 − 0.169i)4-s + (−0.828 − 0.559i)5-s + (0.372 − 0.927i)6-s + (0.873 + 0.487i)7-s + (−0.967 + 0.251i)8-s + (−0.594 − 0.803i)9-s + (0.873 + 0.487i)10-s + (0.985 + 0.169i)11-s + (−0.292 + 0.956i)12-s + (0.210 − 0.977i)13-s + (−0.911 − 0.411i)14-s + (0.873 − 0.487i)15-s + (0.942 − 0.333i)16-s + (−0.450 + 0.892i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.580 + 0.814i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.580 + 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5304062275 + 0.2734416131i\)
\(L(\frac12)\) \(\approx\) \(0.5304062275 + 0.2734416131i\)
\(L(1)\) \(\approx\) \(0.5962167385 + 0.1719754171i\)
\(L(1)\) \(\approx\) \(0.5962167385 + 0.1719754171i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.996 + 0.0848i)T \)
3 \( 1 + (-0.450 + 0.892i)T \)
5 \( 1 + (-0.828 - 0.559i)T \)
7 \( 1 + (0.873 + 0.487i)T \)
11 \( 1 + (0.985 + 0.169i)T \)
13 \( 1 + (0.210 - 0.977i)T \)
17 \( 1 + (-0.450 + 0.892i)T \)
19 \( 1 + (0.660 - 0.750i)T \)
23 \( 1 + (0.210 + 0.977i)T \)
29 \( 1 + (-0.721 + 0.691i)T \)
31 \( 1 + (0.210 + 0.977i)T \)
37 \( 1 + (0.985 + 0.169i)T \)
41 \( 1 + (0.942 - 0.333i)T \)
43 \( 1 + (0.0424 + 0.999i)T \)
47 \( 1 + (0.660 + 0.750i)T \)
53 \( 1 + (0.0424 - 0.999i)T \)
59 \( 1 + (-0.127 - 0.991i)T \)
61 \( 1 + (-0.996 + 0.0848i)T \)
67 \( 1 + (-0.911 + 0.411i)T \)
71 \( 1 + (-0.828 - 0.559i)T \)
73 \( 1 + (0.524 - 0.851i)T \)
79 \( 1 + (0.524 + 0.851i)T \)
83 \( 1 + (0.524 - 0.851i)T \)
89 \( 1 + (0.778 - 0.628i)T \)
97 \( 1 + (0.0424 - 0.999i)T \)
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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

−27.8844664375054023853872882563, −27.0395369078415714428488942902, −26.31880783345384802704315141856, −24.821671643451448294781777493004, −24.32549405785386262071109865641, −23.255557027237895883336411925248, −22.19867508408045279750500040293, −20.63389405702668436036843868794, −19.77073349180708275980344412988, −18.734279750766274174038423979169, −18.26282648830568449431225091298, −17.017957456692500188084845480697, −16.37100698458344015972355446814, −14.84118158918162647574188243540, −13.842273302014795020414705081858, −11.9806379047846670849877635024, −11.54791552114373899415742751574, −10.72060311585134995445758191241, −9.05758325267417024676012051596, −7.83724580095365960362965201553, −7.180722036129912290605719242718, −6.177676484571305483933013483419, −4.140074915512178834495376439526, −2.34332951174947892119950891371, −0.95422900880389767147815041003, 1.22348371217601972082482078442, 3.34342535433604821912799513746, 4.78778216298590091290443854321, 5.95953751937440664192934748418, 7.55926369033649807602821565752, 8.67567027646132236021316080081, 9.37446526643948066735593576045, 10.88794852112787170581319837444, 11.45099045385562715838434801193, 12.44851229028835989959691636402, 14.75570941400123012204106473428, 15.37923002705433019008747846962, 16.24091317951677079330747755920, 17.35992835010791435800296286844, 17.90483112068823247119419954590, 19.53463230348236324354004207222, 20.180439653169309214358506552501, 21.15496802707293159229753503482, 22.21642077127887771808002483388, 23.577581942681178587936361093217, 24.430073887773652810465117227016, 25.4589482249739963136117880738, 26.72588606480753436529895156288, 27.49591520125796773456917035032, 27.93023287810796421104540688571

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