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Properties

Label 1-149-149.25-r0-0-0
Degree $1$
Conductor $149$
Sign $-0.997 - 0.0680i$
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  + (−0.828 + 0.559i)2-s + (−0.127 + 0.991i)3-s + (0.372 − 0.927i)4-s + (0.524 + 0.851i)5-s + (−0.450 − 0.892i)6-s + (−0.911 − 0.411i)7-s + (0.210 + 0.977i)8-s + (−0.967 − 0.251i)9-s + (−0.911 − 0.411i)10-s + (0.372 + 0.927i)11-s + (0.873 + 0.487i)12-s + (−0.996 + 0.0848i)13-s + (0.985 − 0.169i)14-s + (−0.911 + 0.411i)15-s + (−0.721 − 0.691i)16-s + (−0.127 + 0.991i)17-s + ⋯
L(s)  = 1  + (−0.828 + 0.559i)2-s + (−0.127 + 0.991i)3-s + (0.372 − 0.927i)4-s + (0.524 + 0.851i)5-s + (−0.450 − 0.892i)6-s + (−0.911 − 0.411i)7-s + (0.210 + 0.977i)8-s + (−0.967 − 0.251i)9-s + (−0.911 − 0.411i)10-s + (0.372 + 0.927i)11-s + (0.873 + 0.487i)12-s + (−0.996 + 0.0848i)13-s + (0.985 − 0.169i)14-s + (−0.911 + 0.411i)15-s + (−0.721 − 0.691i)16-s + (−0.127 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $-0.997 - 0.0680i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ -0.997 - 0.0680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01689915393 + 0.4964364166i\)
\(L(\frac12)\) \(\approx\) \(0.01689915393 + 0.4964364166i\)
\(L(1)\) \(\approx\) \(0.4130165499 + 0.4314988887i\)
\(L(1)\) \(\approx\) \(0.4130165499 + 0.4314988887i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.828 + 0.559i)T \)
3 \( 1 + (-0.127 + 0.991i)T \)
5 \( 1 + (0.524 + 0.851i)T \)
7 \( 1 + (-0.911 - 0.411i)T \)
11 \( 1 + (0.372 + 0.927i)T \)
13 \( 1 + (-0.996 + 0.0848i)T \)
17 \( 1 + (-0.127 + 0.991i)T \)
19 \( 1 + (0.942 + 0.333i)T \)
23 \( 1 + (-0.996 - 0.0848i)T \)
29 \( 1 + (-0.594 - 0.803i)T \)
31 \( 1 + (-0.996 - 0.0848i)T \)
37 \( 1 + (0.372 + 0.927i)T \)
41 \( 1 + (-0.721 - 0.691i)T \)
43 \( 1 + (-0.292 - 0.956i)T \)
47 \( 1 + (0.942 - 0.333i)T \)
53 \( 1 + (-0.292 + 0.956i)T \)
59 \( 1 + (0.778 + 0.628i)T \)
61 \( 1 + (-0.828 + 0.559i)T \)
67 \( 1 + (0.985 + 0.169i)T \)
71 \( 1 + (0.524 + 0.851i)T \)
73 \( 1 + (0.660 - 0.750i)T \)
79 \( 1 + (0.660 + 0.750i)T \)
83 \( 1 + (0.660 - 0.750i)T \)
89 \( 1 + (0.0424 + 0.999i)T \)
97 \( 1 + (-0.292 + 0.956i)T \)
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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

−27.903734361125744426445752353912, −26.678898580603056288464939873409, −25.53106484188477420064815998675, −24.83010755396435781876785675476, −24.12207842849669926396308545414, −22.37174264042443127086468102800, −21.747534563366332839273346472044, −20.14990087714356415694830871837, −19.74616073378699327786848847828, −18.58430519235406913996639426886, −17.84727882520023523311325429199, −16.694581496785198230244736786543, −16.14368760953811883370681707816, −13.99890865197257932809575520303, −12.987509527263774106037094670750, −12.25301627976434653414556071795, −11.339965874008480293692169159960, −9.6555846171180217698868328909, −9.02973680029690299748096175671, −7.802757564236116656676699070959, −6.637981588883079839917618324395, −5.396967550661045588854252229572, −3.199424654322675284200620576655, −2.02317387882098694469651449924, −0.538646805434432597910364230669, 2.24683920052277172459751364222, 3.85312902802916889836647037813, 5.49360584788918726268601397911, 6.51357043092475300468807402410, 7.55782162846823450149457870842, 9.28370776385197421640079821555, 9.95028142732075935361305434572, 10.51393918147069272107904560122, 11.94338523987389718932674904270, 13.85675954629244028408842272971, 14.82857088500458530164267735394, 15.520514698064699961305003845523, 16.81499644763739830200534964367, 17.29956027800699965384774425544, 18.514107757241939982781889725238, 19.71095527607268208495491404547, 20.41596519469214482893407996259, 22.05894805658334810915791547331, 22.52404973197809849157595416960, 23.70149787695924230889415945362, 25.193308931703602452409474465528, 25.95490258604346748372328012211, 26.52603188542020660999958226267, 27.38728041943205146712570566677, 28.561598784582096629089475190711

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