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Properties

Label 1-151-151.21-r0-0-0
Degree $1$
Conductor $151$
Sign $0.918 + 0.395i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.913 + 0.406i)2-s + (0.876 − 0.481i)3-s + (0.669 + 0.743i)4-s + (−0.999 − 0.0418i)5-s + (0.996 − 0.0836i)6-s + (0.783 + 0.621i)7-s + (0.309 + 0.951i)8-s + (0.535 − 0.844i)9-s + (−0.895 − 0.444i)10-s + (−0.0209 − 0.999i)11-s + (0.944 + 0.328i)12-s + (−0.957 + 0.289i)13-s + (0.463 + 0.886i)14-s + (−0.895 + 0.444i)15-s + (−0.104 + 0.994i)16-s + (0.146 + 0.989i)17-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (0.876 − 0.481i)3-s + (0.669 + 0.743i)4-s + (−0.999 − 0.0418i)5-s + (0.996 − 0.0836i)6-s + (0.783 + 0.621i)7-s + (0.309 + 0.951i)8-s + (0.535 − 0.844i)9-s + (−0.895 − 0.444i)10-s + (−0.0209 − 0.999i)11-s + (0.944 + 0.328i)12-s + (−0.957 + 0.289i)13-s + (0.463 + 0.886i)14-s + (−0.895 + 0.444i)15-s + (−0.104 + 0.994i)16-s + (0.146 + 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.918 + 0.395i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.918 + 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.112607156 + 0.4360876408i\)
\(L(\frac12)\) \(\approx\) \(2.112607156 + 0.4360876408i\)
\(L(1)\) \(\approx\) \(1.914505563 + 0.2913927784i\)
\(L(1)\) \(\approx\) \(1.914505563 + 0.2913927784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.913 + 0.406i)T \)
3 \( 1 + (0.876 - 0.481i)T \)
5 \( 1 + (-0.999 - 0.0418i)T \)
7 \( 1 + (0.783 + 0.621i)T \)
11 \( 1 + (-0.0209 - 0.999i)T \)
13 \( 1 + (-0.957 + 0.289i)T \)
17 \( 1 + (0.146 + 0.989i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.978 + 0.207i)T \)
29 \( 1 + (-0.992 + 0.125i)T \)
31 \( 1 + (-0.699 - 0.714i)T \)
37 \( 1 + (0.228 - 0.973i)T \)
41 \( 1 + (-0.425 + 0.904i)T \)
43 \( 1 + (0.783 - 0.621i)T \)
47 \( 1 + (-0.570 - 0.821i)T \)
53 \( 1 + (-0.187 + 0.982i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.855 + 0.518i)T \)
67 \( 1 + (0.535 + 0.844i)T \)
71 \( 1 + (0.146 - 0.989i)T \)
73 \( 1 + (-0.929 + 0.368i)T \)
79 \( 1 + (0.968 + 0.248i)T \)
83 \( 1 + (-0.637 + 0.770i)T \)
89 \( 1 + (-0.348 - 0.937i)T \)
97 \( 1 + (0.463 - 0.886i)T \)
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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

−27.58908648759941667627072483533, −27.40166225844456785544632054240, −26.045940172605253130574036679342, −24.79677358576004883706187530504, −24.075264789170827157765043449331, −22.92492571963969066651467372969, −22.18547156137191657985437744009, −20.74200573145185054134687343315, −20.34589601799937826911357291323, −19.59346573902963509563887433103, −18.37764708049725792061389966958, −16.5635812664835340310979151775, −15.54378675690634296640090875017, −14.65361299319328174805653932956, −14.12521184886893513060800568510, −12.6917471467812598959707653375, −11.73916638935593358363390149341, −10.55932566581015290058586556162, −9.63926737664828053343765592109, −7.82121011674208921274695219366, −7.260317529239701249111445287615, −5.01742904046896439216185230039, −4.31077947761715170523849630659, −3.254564350947897638193314296211, −1.86325944021047101547837244782, 2.07869930472490400083201004217, 3.332355839851052998092480761779, 4.40048394591625404571896346256, 5.830973217504080653461150519339, 7.31318058093805632480142829990, 8.004911413946121321198792628953, 8.94852408491534306660579829936, 11.15311556172506625482257683198, 12.007613435539479508466995934030, 12.927007278993667303389886123451, 14.14658735451961798094058547352, 14.89654037607835674449020850977, 15.621815851934695582500357768432, 16.8835552574953925941080176918, 18.30568955901871773086736877982, 19.44480952396392034452380298124, 20.19048757085605040279791279369, 21.4058672372748545369284740446, 22.09672292048172197418324165829, 23.78645170947261372311145870325, 24.07062399728745354116696926329, 24.801819050439532557547942922210, 26.14537607457080179005116420077, 26.758871331955156996906780790763, 28.06964952539153868543207396761

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