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Properties

Label 1-149-149.49-r0-0-0
Degree $1$
Conductor $149$
Sign $0.0605 - 0.998i$
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  + (0.873 − 0.487i)2-s + (0.942 − 0.333i)3-s + (0.524 − 0.851i)4-s + (−0.911 − 0.411i)5-s + (0.660 − 0.750i)6-s + (−0.996 + 0.0848i)7-s + (0.0424 − 0.999i)8-s + (0.778 − 0.628i)9-s + (−0.996 + 0.0848i)10-s + (0.524 + 0.851i)11-s + (0.210 − 0.977i)12-s + (−0.292 − 0.956i)13-s + (−0.828 + 0.559i)14-s + (−0.996 − 0.0848i)15-s + (−0.450 − 0.892i)16-s + (0.942 − 0.333i)17-s + ⋯
L(s)  = 1  + (0.873 − 0.487i)2-s + (0.942 − 0.333i)3-s + (0.524 − 0.851i)4-s + (−0.911 − 0.411i)5-s + (0.660 − 0.750i)6-s + (−0.996 + 0.0848i)7-s + (0.0424 − 0.999i)8-s + (0.778 − 0.628i)9-s + (−0.996 + 0.0848i)10-s + (0.524 + 0.851i)11-s + (0.210 − 0.977i)12-s + (−0.292 − 0.956i)13-s + (−0.828 + 0.559i)14-s + (−0.996 − 0.0848i)15-s + (−0.450 − 0.892i)16-s + (0.942 − 0.333i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0605 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0605 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.0605 - 0.998i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.0605 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.425928250 - 1.342040154i\)
\(L(\frac12)\) \(\approx\) \(1.425928250 - 1.342040154i\)
\(L(1)\) \(\approx\) \(1.559479981 - 0.8753465988i\)
\(L(1)\) \(\approx\) \(1.559479981 - 0.8753465988i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.873 - 0.487i)T \)
3 \( 1 + (0.942 - 0.333i)T \)
5 \( 1 + (-0.911 - 0.411i)T \)
7 \( 1 + (-0.996 + 0.0848i)T \)
11 \( 1 + (0.524 + 0.851i)T \)
13 \( 1 + (-0.292 - 0.956i)T \)
17 \( 1 + (0.942 - 0.333i)T \)
19 \( 1 + (0.372 + 0.927i)T \)
23 \( 1 + (-0.292 + 0.956i)T \)
29 \( 1 + (-0.127 + 0.991i)T \)
31 \( 1 + (-0.292 + 0.956i)T \)
37 \( 1 + (0.524 + 0.851i)T \)
41 \( 1 + (-0.450 - 0.892i)T \)
43 \( 1 + (-0.967 + 0.251i)T \)
47 \( 1 + (0.372 - 0.927i)T \)
53 \( 1 + (-0.967 - 0.251i)T \)
59 \( 1 + (-0.721 + 0.691i)T \)
61 \( 1 + (0.873 - 0.487i)T \)
67 \( 1 + (-0.828 - 0.559i)T \)
71 \( 1 + (-0.911 - 0.411i)T \)
73 \( 1 + (0.985 + 0.169i)T \)
79 \( 1 + (0.985 - 0.169i)T \)
83 \( 1 + (0.985 + 0.169i)T \)
89 \( 1 + (-0.594 + 0.803i)T \)
97 \( 1 + (-0.967 - 0.251i)T \)
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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

−28.336637212865506890370479128640, −26.66931038500829162006685253747, −26.51820914681488067085953940687, −25.42853778139476941980942847712, −24.37509784288912818180695688158, −23.527959746407470906653060936047, −22.30984269890577239759233376711, −21.75361602531577278824719772689, −20.47687359855769495859544450594, −19.49425664848678156090047185458, −18.83215561689644162921071902199, −16.663918884878777414158316214265, −16.13426740195692138405049449181, −15.09283770869684450688250129089, −14.289552823838103294048366239, −13.36999236135639215347057956851, −12.1896620518722849045983572167, −11.07181640458850363433301119182, −9.51319555199700229090480010419, −8.288273117347675072118962160489, −7.265459454515755846733828761108, −6.24670317366150527471212953864, −4.39878463396866509921688256928, −3.58609329652211225553667079455, −2.66968679593302952686750602547, 1.41707605125169277577581197765, 3.1541724503546463662540907391, 3.70646747552936026763830242006, 5.218202690103608267176943042554, 6.83157411151574501335770337350, 7.78025810213145143084665532477, 9.35043561618184534234732077678, 10.21624308433537346606963570593, 12.1307762080429056756631450601, 12.40094156804643627244552589450, 13.50637537100154692668664145630, 14.69838750581168679355911333839, 15.42508566055214007093580028249, 16.41076667482099703951937213377, 18.39025232880319287473317097309, 19.39711399433930317461321857651, 20.02445576959749754254510757629, 20.621252573520811938948684682388, 22.055020546940983808929701850723, 23.04019479365538204233988848411, 23.75026054401993323246853443915, 25.1202955594698242813363161250, 25.38139645907831486487047322826, 27.11117089598706876170403700208, 27.90057498426095385128032005369

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