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Properties

Label 2-520-65.64-c1-0-6
Degree $2$
Conductor $520$
Sign $0.339 - 0.940i$
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  + 0.949i·3-s + (0.108 + 2.23i)5-s + 3.85·7-s + 2.09·9-s + 0.589i·11-s + (−3.32 − 1.38i)13-s + (−2.11 + 0.103i)15-s − 3.66i·17-s + 5.94i·19-s + 3.66i·21-s − 3.51i·23-s + (−4.97 + 0.486i)25-s + 4.83i·27-s + 5.33·29-s + 8.71i·31-s + ⋯
L(s)  = 1  + 0.547i·3-s + (0.0486 + 0.998i)5-s + 1.45·7-s + 0.699·9-s + 0.177i·11-s + (−0.922 − 0.384i)13-s + (−0.547 + 0.0266i)15-s − 0.887i·17-s + 1.36i·19-s + 0.798i·21-s − 0.733i·23-s + (−0.995 + 0.0972i)25-s + 0.931i·27-s + 0.991·29-s + 1.56i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.339 - 0.940i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.339 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38091 + 0.969714i\)
\(L(\frac12)\) \(\approx\) \(1.38091 + 0.969714i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.108 - 2.23i)T \)
13 \( 1 + (3.32 + 1.38i)T \)
good3 \( 1 - 0.949iT - 3T^{2} \)
7 \( 1 - 3.85T + 7T^{2} \)
11 \( 1 - 0.589iT - 11T^{2} \)
17 \( 1 + 3.66iT - 17T^{2} \)
19 \( 1 - 5.94iT - 19T^{2} \)
23 \( 1 + 3.51iT - 23T^{2} \)
29 \( 1 - 5.33T + 29T^{2} \)
31 \( 1 - 8.71iT - 31T^{2} \)
37 \( 1 - 1.85T + 37T^{2} \)
41 \( 1 + 4.63iT - 41T^{2} \)
43 \( 1 - 6.30iT - 43T^{2} \)
47 \( 1 + 3.85T + 47T^{2} \)
53 \( 1 + 10.7iT - 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 4.09T + 67T^{2} \)
71 \( 1 + 5.34iT - 71T^{2} \)
73 \( 1 + 6.09T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 8.77T + 83T^{2} \)
89 \( 1 + 0.413iT - 89T^{2} \)
97 \( 1 - 8.45T + 97T^{2} \)
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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

−10.79484202483332036838094443507, −10.32064933489788567229310230645, −9.527483149827050767185748527858, −8.195634715091332776783018552614, −7.51257626492250525455979936727, −6.59035906533841856481531151414, −5.16201083549729032857238510826, −4.51551307780735912165671476192, −3.18036561778596103157398479529, −1.81703855688435847151878889110, 1.16150273466128451053173243839, 2.20522955088067303866698340670, 4.33603205946178041751386259962, 4.82424531515548926935460110534, 6.02431663785123247985879937944, 7.34099171486362303964264539138, 7.918022566865765419680760996077, 8.824372908754242934772038240973, 9.696993417572490898874993282727, 10.82430731863923041556955232527

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