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Properties

Label 2-290-145.108-c1-0-7
Degree $2$
Conductor $290$
Sign $-0.324 - 0.945i$
Analytic cond. $2.31566$
Root an. cond. $1.52172$
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  + (0.433 + 0.900i)2-s + (1.62 + 2.03i)3-s + (−0.623 + 0.781i)4-s + (2.22 − 0.223i)5-s + (−1.12 + 2.34i)6-s + (−3.98 − 0.448i)7-s + (−0.974 − 0.222i)8-s + (−0.834 + 3.65i)9-s + (1.16 + 1.90i)10-s + (0.227 + 0.142i)11-s − 2.59·12-s + (3.20 + 2.01i)13-s + (−1.32 − 3.78i)14-s + (4.05 + 4.15i)15-s + (−0.222 − 0.974i)16-s − 4.78i·17-s + ⋯
L(s)  = 1  + (0.306 + 0.637i)2-s + (0.935 + 1.17i)3-s + (−0.311 + 0.390i)4-s + (0.994 − 0.100i)5-s + (−0.460 + 0.955i)6-s + (−1.50 − 0.169i)7-s + (−0.344 − 0.0786i)8-s + (−0.278 + 1.21i)9-s + (0.369 + 0.603i)10-s + (0.0685 + 0.0430i)11-s − 0.750·12-s + (0.889 + 0.558i)13-s + (−0.353 − 1.01i)14-s + (1.04 + 1.07i)15-s + (−0.0556 − 0.243i)16-s − 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(290\)    =    \(2 \cdot 5 \cdot 29\)
Sign: $-0.324 - 0.945i$
Analytic conductor: \(2.31566\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{290} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 290,\ (\ :1/2),\ -0.324 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13767 + 1.59303i\)
\(L(\frac12)\) \(\approx\) \(1.13767 + 1.59303i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.433 - 0.900i)T \)
5 \( 1 + (-2.22 + 0.223i)T \)
29 \( 1 + (0.888 - 5.31i)T \)
good3 \( 1 + (-1.62 - 2.03i)T + (-0.667 + 2.92i)T^{2} \)
7 \( 1 + (3.98 + 0.448i)T + (6.82 + 1.55i)T^{2} \)
11 \( 1 + (-0.227 - 0.142i)T + (4.77 + 9.91i)T^{2} \)
13 \( 1 + (-3.20 - 2.01i)T + (5.64 + 11.7i)T^{2} \)
17 \( 1 + 4.78iT - 17T^{2} \)
19 \( 1 + (-1.20 + 0.136i)T + (18.5 - 4.22i)T^{2} \)
23 \( 1 + (2.15 + 6.17i)T + (-17.9 + 14.3i)T^{2} \)
31 \( 1 + (1.94 - 5.57i)T + (-24.2 - 19.3i)T^{2} \)
37 \( 1 + (-1.89 + 8.29i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-4.83 - 4.83i)T + 41iT^{2} \)
43 \( 1 + (6.44 + 3.10i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (0.460 + 2.01i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-2.95 - 1.03i)T + (41.4 + 33.0i)T^{2} \)
59 \( 1 + 6.00iT - 59T^{2} \)
61 \( 1 + (13.6 + 1.53i)T + (59.4 + 13.5i)T^{2} \)
67 \( 1 + (-9.25 + 5.81i)T + (29.0 - 60.3i)T^{2} \)
71 \( 1 + (9.50 - 2.17i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (4.68 - 9.73i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + (0.719 - 0.452i)T + (34.2 - 71.1i)T^{2} \)
83 \( 1 + (8.11 - 0.914i)T + (80.9 - 18.4i)T^{2} \)
89 \( 1 + (-14.8 - 5.20i)T + (69.5 + 55.4i)T^{2} \)
97 \( 1 + (0.902 - 1.13i)T + (-21.5 - 94.5i)T^{2} \)
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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

−12.44670288763689489554966330279, −10.75804005337850730696575036344, −9.854144525871145474349624299560, −9.230204844416396053390674655514, −8.668415913435687271993694880304, −7.01099840909137639370784569880, −6.15609275007634326174053959535, −4.91566339294908250240787526623, −3.72693883176196754876689810260, −2.79345064939460744449082014779, 1.50804816922740720840803093296, 2.73594379568690953033297583012, 3.62208005670547721154065973489, 5.90061006061066399931260207622, 6.31420754755045626331737806378, 7.68209187413780908552657632285, 8.824530334589006504961832883387, 9.624231171081991758395770482248, 10.41686940476473388640620439768, 11.81101568773127869569739029265

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