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Properties

Label 2-520-65.57-c1-0-20
Degree $2$
Conductor $520$
Sign $-0.617 + 0.786i$
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 + (−2.07 − 0.831i)5-s − 4.71i·7-s − 3.18i·9-s + (−0.281 − 0.281i)11-s + (1.52 + 3.26i)13-s + (−5.11 + 2.18i)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 + (−0.928 − 0.371i)5-s − 1.78i·7-s − 1.06i·9-s + (−0.0847 − 0.0847i)11-s + (0.424 + 0.905i)13-s + (−1.32 + 0.565i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.617 + 0.786i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.673114 - 1.38490i\)
\(L(\frac12)\) \(\approx\) \(0.673114 - 1.38490i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 + 0.831i)T \)
13 \( 1 + (-1.52 - 3.26i)T \)
good3 \( 1 + (-1.75 + 1.75i)T - 3iT^{2} \)
7 \( 1 + 4.71iT - 7T^{2} \)
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 \( 1 + 3.15iT - 37T^{2} \)
41 \( 1 + (-8.37 + 8.37i)T - 41iT^{2} \)
43 \( 1 + (-3.55 - 3.55i)T + 43iT^{2} \)
47 \( 1 - 3.57iT - 47T^{2} \)
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 \( 1 - 7.84T + 67T^{2} \)
71 \( 1 + (5.41 - 5.41i)T - 71iT^{2} \)
73 \( 1 - 9.61T + 73T^{2} \)
79 \( 1 + 1.86iT - 79T^{2} \)
83 \( 1 - 6.71iT - 83T^{2} \)
89 \( 1 + (10.0 - 10.0i)T - 89iT^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−10.74454847969232528844403703865, −9.479723572812296214317648933385, −8.383211261795852913339062427026, −7.960044074737112981661274909353, −7.09999532529449123003278819298, −6.40594318257995970133276122119, −4.16158608221375508342443561519, −4.03464410395975189260250379294, −2.26627121335985450779578422199, −0.828141773707851724601097698449, 2.61660636851593743469590914011, 3.15458018677732701086542192554, 4.41120526806486792741953848076, 5.36307142995992507302285279753, 6.67514168337689687124219131665, 8.016498986910266252761850165350, 8.620563105113621831642347571868, 9.213722504379501878655953105133, 10.19215290682268248059571347529, 11.15168819686671548262656656272

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