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Properties

Label 2-151-151.59-c1-0-4
Degree $2$
Conductor $151$
Sign $0.224 + 0.974i$
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  − 1.66·2-s + (−0.661 − 0.480i)3-s + 0.764·4-s + (0.441 + 1.35i)5-s + (1.09 + 0.798i)6-s + (−0.446 − 1.37i)7-s + 2.05·8-s + (−0.720 − 2.21i)9-s + (−0.733 − 2.25i)10-s + (2.53 − 1.84i)11-s + (−0.505 − 0.367i)12-s + (−3.51 − 2.55i)13-s + (0.741 + 2.28i)14-s + (0.360 − 1.10i)15-s − 4.94·16-s + (2.29 − 7.06i)17-s + ⋯
L(s)  = 1  − 1.17·2-s + (−0.381 − 0.277i)3-s + 0.382·4-s + (0.197 + 0.607i)5-s + (0.448 + 0.326i)6-s + (−0.168 − 0.518i)7-s + 0.726·8-s + (−0.240 − 0.739i)9-s + (−0.231 − 0.713i)10-s + (0.763 − 0.554i)11-s + (−0.145 − 0.105i)12-s + (−0.973 − 0.707i)13-s + (0.198 + 0.610i)14-s + (0.0930 − 0.286i)15-s − 1.23·16-s + (0.557 − 1.71i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(151\)
Sign: $0.224 + 0.974i$
Analytic conductor: \(1.20574\)
Root analytic conductor: \(1.09806\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 151,\ (\ :1/2),\ 0.224 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378035 - 0.300996i\)
\(L(\frac12)\) \(\approx\) \(0.378035 - 0.300996i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 + (-11.4 + 4.42i)T \)
good2 \( 1 + 1.66T + 2T^{2} \)
3 \( 1 + (0.661 + 0.480i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.441 - 1.35i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.446 + 1.37i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-2.53 + 1.84i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.51 + 2.55i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.29 + 7.06i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 - 2.05T + 19T^{2} \)
23 \( 1 - 3.41T + 23T^{2} \)
29 \( 1 + (1.08 - 0.788i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.77 - 8.53i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (9.15 + 6.65i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.38 - 4.64i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (0.632 - 1.94i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (2.74 + 1.99i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.10 - 1.53i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 - 0.262T + 59T^{2} \)
61 \( 1 + (-4.59 + 3.33i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.517 + 1.59i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.55 - 7.87i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.430 + 1.32i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.70 + 14.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.89 - 5.83i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (0.756 - 2.32i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.77 - 5.46i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.58812250444448677478020789335, −11.52528440574297949430009506979, −10.59435145296370668261960489177, −9.653889056656049157500612226560, −8.875364984147072225278192611590, −7.35788639420263106934866398235, −6.84638990503498057348524675810, −5.20629942404145207725957372612, −3.19126413805423566429820135855, −0.77482850337215994989415311944, 1.81094230807888501766406494824, 4.39092719951506151449924367671, 5.55037953465677095903816414758, 7.12888027565101919545228609296, 8.290458140900290515423230606499, 9.218872366853892534270050341414, 9.914677341545129694103978324179, 10.95602058681292853316523885726, 12.05373526684827472016370559352, 13.03256598755309949924549280894

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