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Properties

Label 2-151-151.32-c1-0-8
Degree $2$
Conductor $151$
Sign $0.999 - 0.00632i$
Analytic cond. $1.20574$
Root an. cond. $1.09806$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.368 + 0.637i)2-s + 1.32·3-s + (0.729 − 1.26i)4-s + (−0.697 − 1.20i)5-s + (0.488 + 0.846i)6-s + (−0.120 − 0.208i)7-s + 2.54·8-s − 1.23·9-s + (0.513 − 0.888i)10-s + (0.0188 + 0.0325i)11-s + (0.968 − 1.67i)12-s + (−3.10 + 5.37i)13-s + (0.0884 − 0.153i)14-s + (−0.925 − 1.60i)15-s + (−0.521 − 0.903i)16-s + (2.51 + 4.35i)17-s + ⋯
L(s)  = 1  + (0.260 + 0.450i)2-s + 0.766·3-s + (0.364 − 0.631i)4-s + (−0.311 − 0.540i)5-s + (0.199 + 0.345i)6-s + (−0.0454 − 0.0786i)7-s + 0.899·8-s − 0.412·9-s + (0.162 − 0.281i)10-s + (0.00567 + 0.00982i)11-s + (0.279 − 0.484i)12-s + (−0.861 + 1.49i)13-s + (0.0236 − 0.0409i)14-s + (−0.239 − 0.413i)15-s + (−0.130 − 0.225i)16-s + (0.610 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(151\)
Sign: $0.999 - 0.00632i$
Analytic conductor: \(1.20574\)
Root analytic conductor: \(1.09806\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 151,\ (\ :1/2),\ 0.999 - 0.00632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56340 + 0.00494502i\)
\(L(\frac12)\) \(\approx\) \(1.56340 + 0.00494502i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 + (-8.34 + 9.02i)T \)
good2 \( 1 + (-0.368 - 0.637i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 - 1.32T + 3T^{2} \)
5 \( 1 + (0.697 + 1.20i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.120 + 0.208i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0188 - 0.0325i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.10 - 5.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.51 - 4.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 3.37T + 19T^{2} \)
23 \( 1 + (-3.23 - 5.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.31T + 29T^{2} \)
31 \( 1 + (2.94 + 5.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.07 + 8.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.953T + 41T^{2} \)
43 \( 1 + (1.56 - 2.71i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.791 + 1.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.65T + 53T^{2} \)
59 \( 1 - 6.45T + 59T^{2} \)
61 \( 1 + (-1.01 + 1.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 9.75T + 67T^{2} \)
71 \( 1 + (1.85 - 3.21i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.595T + 73T^{2} \)
79 \( 1 + 4.91T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + (1.10 - 1.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.973 - 1.68i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.24623947078842730843867282068, −12.06338077704152656500311925523, −11.02805470070834942238213215944, −9.775814523413075150918718240238, −8.810881593922241082619545157135, −7.71793139837306587425531781252, −6.60290141112856658927298021306, −5.30322806652624168696655426820, −4.01276333447174061847568792917, −2.04151748115406556672760234926, 2.72316832803007459640769412194, 3.24009019618344745695715528388, 5.00083700379438673235845057843, 6.87388423210666959813871889658, 7.81949561409357357406761956045, 8.676446247994291696057824508790, 10.18158074480307063100977315277, 11.03904899137676828371834848574, 12.13969057825981084459095844056, 12.85309571697177496956498775600

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