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Properties

Label 2-520-65.18-c1-0-2
Degree $2$
Conductor $520$
Sign $-0.530 - 0.847i$
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  + (1.75 + 1.75i)3-s + (0.831 + 2.07i)5-s − 4.71·7-s + 3.18i·9-s + (−0.281 − 0.281i)11-s + (−3.26 + 1.52i)13-s + (−2.18 + 5.11i)15-s + (4.84 + 4.84i)17-s + (0.782 + 0.782i)19-s + (−8.29 − 8.29i)21-s + (2.30 − 2.30i)23-s + (−3.61 + 3.45i)25-s + (−0.331 + 0.331i)27-s + 6.24i·29-s + (6.66 − 6.66i)31-s + ⋯
L(s)  = 1  + (1.01 + 1.01i)3-s + (0.371 + 0.928i)5-s − 1.78·7-s + 1.06i·9-s + (−0.0847 − 0.0847i)11-s + (−0.905 + 0.424i)13-s + (−0.565 + 1.32i)15-s + (1.17 + 1.17i)17-s + (0.179 + 0.179i)19-s + (−1.80 − 1.80i)21-s + (0.480 − 0.480i)23-s + (−0.723 + 0.690i)25-s + (−0.0637 + 0.0637i)27-s + 1.16i·29-s + (1.19 − 1.19i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.530 - 0.847i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.785305 + 1.41824i\)
\(L(\frac12)\) \(\approx\) \(0.785305 + 1.41824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.831 - 2.07i)T \)
13 \( 1 + (3.26 - 1.52i)T \)
good3 \( 1 + (-1.75 - 1.75i)T + 3iT^{2} \)
7 \( 1 + 4.71T + 7T^{2} \)
11 \( 1 + (0.281 + 0.281i)T + 11iT^{2} \)
17 \( 1 + (-4.84 - 4.84i)T + 17iT^{2} \)
19 \( 1 + (-0.782 - 0.782i)T + 19iT^{2} \)
23 \( 1 + (-2.30 + 2.30i)T - 23iT^{2} \)
29 \( 1 - 6.24iT - 29T^{2} \)
31 \( 1 + (-6.66 + 6.66i)T - 31iT^{2} \)
37 \( 1 + 3.15T + 37T^{2} \)
41 \( 1 + (-8.37 + 8.37i)T - 41iT^{2} \)
43 \( 1 + (3.55 - 3.55i)T - 43iT^{2} \)
47 \( 1 - 3.57T + 47T^{2} \)
53 \( 1 + (-1.47 - 1.47i)T + 53iT^{2} \)
59 \( 1 + (4.52 - 4.52i)T - 59iT^{2} \)
61 \( 1 - 4.05T + 61T^{2} \)
67 \( 1 + 7.84iT - 67T^{2} \)
71 \( 1 + (5.41 - 5.41i)T - 71iT^{2} \)
73 \( 1 - 9.61iT - 73T^{2} \)
79 \( 1 - 1.86iT - 79T^{2} \)
83 \( 1 + 6.71T + 83T^{2} \)
89 \( 1 + (-10.0 + 10.0i)T - 89iT^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−10.64843180490013081551766955359, −10.08154751956534093651468797852, −9.618047873915041678397460244278, −8.821361563783313373978257860096, −7.57168300086496196343238464773, −6.61695628292828444849551641418, −5.69391486913644919468139799547, −4.08678564003676195117604163752, −3.25988349492525605862441238439, −2.57879427908126796918087455452, 0.850215014256574181766171726586, 2.55605922517934355370303666830, 3.26757110675021679213329820784, 4.96564790578352704975372559526, 6.10474757141341227146468866288, 7.11603705021661510773903503468, 7.79801119757068060437545420568, 8.856135405710854316629449672458, 9.615199325377981649425638081524, 10.06168299280004504362215166330

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