Properties

Label 2-920-920.29-c1-0-137
Degree $2$
Conductor $920$
Sign $-0.982 - 0.185i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  + (1.08 − 0.903i)2-s + (0.108 + 0.0317i)3-s + (0.365 − 1.96i)4-s + (−0.703 − 2.12i)5-s + (0.146 − 0.0632i)6-s + (−1.73 − 0.250i)7-s + (−1.37 − 2.46i)8-s + (−2.51 − 1.61i)9-s + (−2.68 − 1.67i)10-s + (0.262 − 0.119i)11-s + (0.102 − 0.201i)12-s + (0.641 + 4.45i)13-s + (−2.11 + 1.29i)14-s + (−0.00863 − 0.251i)15-s + (−3.73 − 1.43i)16-s + (−0.884 − 0.766i)17-s + ⋯
L(s)  = 1  + (0.769 − 0.639i)2-s + (0.0624 + 0.0183i)3-s + (0.182 − 0.983i)4-s + (−0.314 − 0.949i)5-s + (0.0597 − 0.0258i)6-s + (−0.657 − 0.0945i)7-s + (−0.487 − 0.873i)8-s + (−0.837 − 0.538i)9-s + (−0.848 − 0.529i)10-s + (0.0790 − 0.0360i)11-s + (0.0294 − 0.0580i)12-s + (0.177 + 1.23i)13-s + (−0.565 + 0.347i)14-s + (−0.00222 − 0.0650i)15-s + (−0.933 − 0.359i)16-s + (−0.214 − 0.185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.982 - 0.185i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ -0.982 - 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.122684 + 1.31032i\)
\(L(\frac12)\) \(\approx\) \(0.122684 + 1.31032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 + 0.903i)T \)
5 \( 1 + (0.703 + 2.12i)T \)
23 \( 1 + (-1.54 + 4.54i)T \)
good3 \( 1 + (-0.108 - 0.0317i)T + (2.52 + 1.62i)T^{2} \)
7 \( 1 + (1.73 + 0.250i)T + (6.71 + 1.97i)T^{2} \)
11 \( 1 + (-0.262 + 0.119i)T + (7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.641 - 4.45i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (0.884 + 0.766i)T + (2.41 + 16.8i)T^{2} \)
19 \( 1 + (1.95 - 1.69i)T + (2.70 - 18.8i)T^{2} \)
29 \( 1 + (2.27 + 1.96i)T + (4.12 + 28.7i)T^{2} \)
31 \( 1 + (-3.13 + 0.921i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-4.34 - 2.79i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (2.35 - 1.51i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-6.63 - 1.94i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + 12.7iT - 47T^{2} \)
53 \( 1 + (-1.29 + 8.97i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (9.72 - 1.39i)T + (56.6 - 16.6i)T^{2} \)
61 \( 1 + (1.96 + 6.69i)T + (-51.3 + 32.9i)T^{2} \)
67 \( 1 + (-1.59 + 3.48i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-5.64 + 12.3i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-3.57 + 3.09i)T + (10.3 - 72.2i)T^{2} \)
79 \( 1 + (2.24 + 15.5i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-7.18 - 4.61i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (11.8 + 3.48i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (-3.37 - 5.24i)T + (-40.2 + 88.2i)T^{2} \)
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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

−9.505262182104471484470680699818, −9.117441933083519264680028504916, −8.147801427796217172230019854296, −6.65304724925058241817289690848, −6.15516423670612155356858179714, −4.99083709493055497321022122163, −4.17065484358307415287311236218, −3.32254254871882516836291963287, −2.01723785933091426416368519808, −0.44900412214504934985063798636, 2.65288812189245587551239107821, 3.16033836433145682080312576904, 4.26819674700193243152080722161, 5.57372278240350328439496285218, 6.08797308346341640130128065766, 7.10040109660439941942423691584, 7.78649419433176852670351376532, 8.564751117924645869709562181862, 9.634272629650897952347020843085, 10.95970734087838753822121215285

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