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Properties

Label 2-520-104.101-c1-0-1
Degree $2$
Conductor $520$
Sign $-5.70e-6 - 0.999i$
Analytic cond. $4.15222$
Root an. cond. $2.03769$
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  + (−0.0643 − 1.41i)2-s + (−0.756 + 0.436i)3-s + (−1.99 + 0.181i)4-s + 5-s + (0.666 + 1.04i)6-s + (−0.0742 − 0.0428i)7-s + (0.384 + 2.80i)8-s + (−1.11 + 1.93i)9-s + (−0.0643 − 1.41i)10-s + (−2.27 − 3.94i)11-s + (1.42 − 1.00i)12-s + (−3.32 + 1.40i)13-s + (−0.0557 + 0.107i)14-s + (−0.756 + 0.436i)15-s + (3.93 − 0.723i)16-s + (−1.52 + 2.64i)17-s + ⋯
L(s)  = 1  + (−0.0454 − 0.998i)2-s + (−0.436 + 0.252i)3-s + (−0.995 + 0.0908i)4-s + 0.447·5-s + (0.271 + 0.425i)6-s + (−0.0280 − 0.0162i)7-s + (0.136 + 0.990i)8-s + (−0.372 + 0.645i)9-s + (−0.0203 − 0.446i)10-s + (−0.686 − 1.18i)11-s + (0.412 − 0.290i)12-s + (−0.921 + 0.389i)13-s + (−0.0149 + 0.0287i)14-s + (−0.195 + 0.112i)15-s + (0.983 − 0.180i)16-s + (−0.370 + 0.641i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-5.70e-6 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-5.70e-6 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-5.70e-6 - 0.999i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -5.70e-6 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255488 + 0.255489i\)
\(L(\frac12)\) \(\approx\) \(0.255488 + 0.255489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0643 + 1.41i)T \)
5 \( 1 - T \)
13 \( 1 + (3.32 - 1.40i)T \)
good3 \( 1 + (0.756 - 0.436i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.0742 + 0.0428i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.27 + 3.94i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.52 - 2.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.10 - 5.38i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.442 + 0.766i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.25 - 4.18i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.361iT - 31T^{2} \)
37 \( 1 + (-5.44 - 9.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.68 - 1.54i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.38 - 4.26i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.90iT - 47T^{2} \)
53 \( 1 - 7.26iT - 53T^{2} \)
59 \( 1 + (-3.94 + 6.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.64 + 4.41i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.77 + 4.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.246 + 0.142i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.37iT - 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + 7.10T + 83T^{2} \)
89 \( 1 + (-10.3 + 5.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.3 - 7.69i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.95040161187000350782132113258, −10.44557394115675517839080694217, −9.614365239058419489396695272201, −8.540953179980151291729044958583, −7.86378200280724023321568309704, −6.13529414210052718237481813937, −5.36226812865950741449413931755, −4.38695937513291501591930919443, −3.05602156960321849492492279869, −1.89314187400328996423565470376, 0.21618086894065761971780166738, 2.48367196396116793301083293658, 4.30017120522233604146732714216, 5.24258133215194095375816288443, 6.02942101877293388489039364799, 7.11445876953455611624378776072, 7.54904331390661115109698509797, 9.000065203844554809423825050650, 9.497974356645470707561550529768, 10.45549819727080374209522507752

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