↑

Properties

Label 1-151-151.125-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.423 - 0.905i$
Analytic cond. $0.701241$
Root an. cond. $0.701241$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.929 + 0.368i)3-s + (−0.809 − 0.587i)4-s + (0.728 − 0.684i)5-s + (0.0627 + 0.998i)6-s + (0.876 + 0.481i)7-s + (−0.809 + 0.587i)8-s + (0.728 − 0.684i)9-s + (−0.425 − 0.904i)10-s + (−0.929 − 0.368i)11-s + (0.968 + 0.248i)12-s + (0.535 − 0.844i)13-s + (0.728 − 0.684i)14-s + (−0.425 + 0.904i)15-s + (0.309 + 0.951i)16-s + (0.876 − 0.481i)17-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.929 + 0.368i)3-s + (−0.809 − 0.587i)4-s + (0.728 − 0.684i)5-s + (0.0627 + 0.998i)6-s + (0.876 + 0.481i)7-s + (−0.809 + 0.587i)8-s + (0.728 − 0.684i)9-s + (−0.425 − 0.904i)10-s + (−0.929 − 0.368i)11-s + (0.968 + 0.248i)12-s + (0.535 − 0.844i)13-s + (0.728 − 0.684i)14-s + (−0.425 + 0.904i)15-s + (0.309 + 0.951i)16-s + (0.876 − 0.481i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.423 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.423 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.423 - 0.905i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.423 - 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5319464718 - 0.8358617892i\)
\(L(\frac12)\) \(\approx\) \(0.5319464718 - 0.8358617892i\)
\(L(1)\) \(\approx\) \(0.7995256705 - 0.5725021424i\)
\(L(1)\) \(\approx\) \(0.7995256705 - 0.5725021424i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (-0.929 + 0.368i)T \)
5 \( 1 + (0.728 - 0.684i)T \)
7 \( 1 + (0.876 + 0.481i)T \)
11 \( 1 + (-0.929 - 0.368i)T \)
13 \( 1 + (0.535 - 0.844i)T \)
17 \( 1 + (0.876 - 0.481i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.637 + 0.770i)T \)
31 \( 1 + (-0.187 - 0.982i)T \)
37 \( 1 + (0.535 - 0.844i)T \)
41 \( 1 + (0.0627 + 0.998i)T \)
43 \( 1 + (0.876 - 0.481i)T \)
47 \( 1 + (0.0627 + 0.998i)T \)
53 \( 1 + (0.968 - 0.248i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.929 - 0.368i)T \)
67 \( 1 + (0.728 + 0.684i)T \)
71 \( 1 + (0.876 + 0.481i)T \)
73 \( 1 + (0.876 + 0.481i)T \)
79 \( 1 + (-0.187 + 0.982i)T \)
83 \( 1 + (-0.992 - 0.125i)T \)
89 \( 1 + (-0.992 - 0.125i)T \)
97 \( 1 + (0.728 + 0.684i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.263243577562739461712469407809, −27.31433654653717668087015811919, −26.18510423635341686689729689078, −25.46049521891919573584675151986, −24.21550464393096115088095306060, −23.491480561606983823247136610138, −22.87310825122750594784094800508, −21.538375328793658487569728343824, −21.12475660642750843181188258965, −18.80325342434219830364801227517, −18.16261033738694960268056349529, −17.31244362884836021109072754238, −16.57340784628269560913679207182, −15.27448828153775438750150367805, −14.15611138900076446569296069903, −13.39858672978288463119441383939, −12.21121707231255449108182502929, −10.8927500525680363038724406008, −9.91660604899096266617755691363, −8.10196607054315876406772291284, −7.20773623999051276790918921975, −6.130205179558663880824552716157, −5.29682749127294158712361354324, −4.01400493597144898498473113443, −1.82665160496469263595539245247, 0.94423526612270620897240741903, 2.45294160792472513301127961411, 4.26111370110742765402440613346, 5.37941722399293873225963329216, 5.798099755717949591135173808538, 8.200711010204174228143809173954, 9.391758630831309440429178538230, 10.48620822297871132098955154462, 11.225860903674240812707211538955, 12.43642513998874043700812507300, 13.10717558790994232689805672359, 14.44872181633610332669955878511, 15.67038346385706482901857619842, 16.93563681127519234963183978368, 18.061179152639014185604791741953, 18.432103443822988883621645619064, 20.29405058639432329966499303339, 21.07745181065005142802248581768, 21.56054257066361766032255176978, 22.637784390516222004234707603262, 23.7275796326796111937188179619, 24.38441843049975420294406211091, 25.92940198442935896050500171500, 27.41028328355382534887200308027, 27.94589834130379090639073373551

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