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Properties

Label 1-149-149.81-r0-0-0
Degree $1$
Conductor $149$
Sign $-0.999 + 0.0210i$
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  + (−0.594 + 0.803i)2-s + (−0.911 − 0.411i)3-s + (−0.292 − 0.956i)4-s + (−0.967 − 0.251i)5-s + (0.873 − 0.487i)6-s + (0.778 − 0.628i)7-s + (0.942 + 0.333i)8-s + (0.660 + 0.750i)9-s + (0.778 − 0.628i)10-s + (−0.292 + 0.956i)11-s + (−0.127 + 0.991i)12-s + (−0.721 − 0.691i)13-s + (0.0424 + 0.999i)14-s + (0.778 + 0.628i)15-s + (−0.828 + 0.559i)16-s + (−0.911 − 0.411i)17-s + ⋯
L(s)  = 1  + (−0.594 + 0.803i)2-s + (−0.911 − 0.411i)3-s + (−0.292 − 0.956i)4-s + (−0.967 − 0.251i)5-s + (0.873 − 0.487i)6-s + (0.778 − 0.628i)7-s + (0.942 + 0.333i)8-s + (0.660 + 0.750i)9-s + (0.778 − 0.628i)10-s + (−0.292 + 0.956i)11-s + (−0.127 + 0.991i)12-s + (−0.721 − 0.691i)13-s + (0.0424 + 0.999i)14-s + (0.778 + 0.628i)15-s + (−0.828 + 0.559i)16-s + (−0.911 − 0.411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $-0.999 + 0.0210i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ -0.999 + 0.0210i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0005748757359 + 0.05468962628i\)
\(L(\frac12)\) \(\approx\) \(0.0005748757359 + 0.05468962628i\)
\(L(1)\) \(\approx\) \(0.3785517480 + 0.06386586401i\)
\(L(1)\) \(\approx\) \(0.3785517480 + 0.06386586401i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.594 + 0.803i)T \)
3 \( 1 + (-0.911 - 0.411i)T \)
5 \( 1 + (-0.967 - 0.251i)T \)
7 \( 1 + (0.778 - 0.628i)T \)
11 \( 1 + (-0.292 + 0.956i)T \)
13 \( 1 + (-0.721 - 0.691i)T \)
17 \( 1 + (-0.911 - 0.411i)T \)
19 \( 1 + (-0.996 - 0.0848i)T \)
23 \( 1 + (-0.721 + 0.691i)T \)
29 \( 1 + (0.524 + 0.851i)T \)
31 \( 1 + (-0.721 + 0.691i)T \)
37 \( 1 + (-0.292 + 0.956i)T \)
41 \( 1 + (-0.828 + 0.559i)T \)
43 \( 1 + (-0.450 - 0.892i)T \)
47 \( 1 + (-0.996 + 0.0848i)T \)
53 \( 1 + (-0.450 + 0.892i)T \)
59 \( 1 + (0.985 + 0.169i)T \)
61 \( 1 + (-0.594 + 0.803i)T \)
67 \( 1 + (0.0424 - 0.999i)T \)
71 \( 1 + (-0.967 - 0.251i)T \)
73 \( 1 + (0.210 + 0.977i)T \)
79 \( 1 + (0.210 - 0.977i)T \)
83 \( 1 + (0.210 + 0.977i)T \)
89 \( 1 + (0.372 - 0.927i)T \)
97 \( 1 + (-0.450 + 0.892i)T \)
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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

−27.74770466495979420796736898089, −26.87236383193911036017853829312, −26.35901711560000865615639993071, −24.4635526805413381481714851374, −23.65466607894657925853272490741, −22.34791347518932776336613316191, −21.69652035830604662496282526357, −20.85753448225239573224234817119, −19.480255835441670375307618546097, −18.69183188965698340616103324657, −17.76504910657917999473975078587, −16.715881515664593859599293915572, −15.809853571690661407975306716364, −14.62167075283420614803416447876, −12.81766944347289524363794521452, −11.77595560790669174281546256592, −11.26122062120392397962319167108, −10.36855287450355909681078893089, −8.897653624404166433292148621661, −7.97596681621385103300306989381, −6.50355571895096220714132849084, −4.77665109448615890559295556211, −3.87316293177624584453105180690, −2.17417526844843060105528589773, −0.06409220895313055208397033946, 1.63094387727206226374891295549, 4.517425488289514984652076104866, 5.104872130538895160192869342056, 6.82922434608691787524127549183, 7.4821909774406875427414510891, 8.41875087933769782416583221104, 10.150658602293297808093556343044, 10.98756129154862588489147934673, 12.14481908022658984416969379916, 13.35019880067401381167219306048, 14.83817322792501305243642775076, 15.65736987193344004552728382049, 16.74310381227240118159373550906, 17.58554892717355437498633060267, 18.22768569033598651230958898792, 19.587555158084214630166232027068, 20.25638347512575796322141339208, 22.124443755176839957362726127510, 23.22552257116181072515327456819, 23.7017666520928186073544482336, 24.47791126585807240258168940246, 25.57586673868285685490443486855, 27.0626610618122124947923174127, 27.4351611645263722667928200121, 28.28979489310673663755502827545

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