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Properties

Label 2-151-151.8-c1-0-3
Degree $2$
Conductor $151$
Sign $0.576 - 0.816i$
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  − 1.97·2-s + (0.893 + 2.74i)3-s + 1.89·4-s + (2.22 − 1.61i)5-s + (−1.76 − 5.42i)6-s + (3.12 − 2.27i)7-s + 0.199·8-s + (−4.33 + 3.14i)9-s + (−4.39 + 3.19i)10-s + (−0.507 + 1.56i)11-s + (1.69 + 5.22i)12-s + (1.02 + 3.15i)13-s + (−6.17 + 4.48i)14-s + (6.43 + 4.67i)15-s − 4.19·16-s + (−6.26 − 4.55i)17-s + ⋯
L(s)  = 1  − 1.39·2-s + (0.515 + 1.58i)3-s + 0.949·4-s + (0.995 − 0.723i)5-s + (−0.720 − 2.21i)6-s + (1.18 − 0.858i)7-s + 0.0704·8-s + (−1.44 + 1.04i)9-s + (−1.38 + 1.00i)10-s + (−0.153 + 0.470i)11-s + (0.489 + 1.50i)12-s + (0.284 + 0.874i)13-s + (−1.64 + 1.19i)14-s + (1.66 + 1.20i)15-s − 1.04·16-s + (−1.51 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(151\)
Sign: $0.576 - 0.816i$
Analytic conductor: \(1.20574\)
Root analytic conductor: \(1.09806\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 151,\ (\ :1/2),\ 0.576 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.730639 + 0.378525i\)
\(L(\frac12)\) \(\approx\) \(0.730639 + 0.378525i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 + (2.41 - 12.0i)T \)
good2 \( 1 + 1.97T + 2T^{2} \)
3 \( 1 + (-0.893 - 2.74i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.22 + 1.61i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-3.12 + 2.27i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.507 - 1.56i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.02 - 3.15i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (6.26 + 4.55i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 - 5.62T + 19T^{2} \)
23 \( 1 + 5.06T + 23T^{2} \)
29 \( 1 + (1.13 - 3.49i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.20 + 1.59i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.0321 - 0.0988i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.749 + 2.30i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (0.531 + 0.385i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (2.81 + 8.66i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.29 - 10.1i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 - 4.26T + 59T^{2} \)
61 \( 1 + (0.587 - 1.80i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (5.26 + 3.82i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-10.6 + 7.76i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.98 + 3.62i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.47 - 2.52i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.01 + 2.91i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.90 + 1.38i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (13.8 + 10.0i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.75780358638860972735023107084, −11.46726355395933717946509408380, −10.71642786658761621017045852510, −9.744628298605537473376164021437, −9.272526598903683298415297803965, −8.491543262797380373708516089185, −7.26732835395237064404461258654, −5.07805510657158311011500406449, −4.32409772860355583557491290448, −1.87461941037437150094368487786, 1.61096690497392450218072325208, 2.46033431304655191086678359907, 5.75597667743618154349661207815, 6.75512886905194133649301838032, 8.070977035757747451226892601846, 8.305868502748702634695120722724, 9.521925834233466807221596384009, 10.79030616216905829356531322261, 11.60519489896591612644066024537, 12.99190259443844012914671670408

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