Properties

Label 2-520-5.4-c1-0-7
Degree $2$
Conductor $520$
Sign $0.498 - 0.866i$
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  + 2.57i·3-s + (1.11 − 1.93i)5-s − 0.632i·7-s − 3.63·9-s + 6.16·11-s + i·13-s + (4.99 + 2.87i)15-s + 4.23i·17-s + 5.31·19-s + 1.62·21-s + 2.34i·23-s + (−2.51 − 4.32i)25-s − 1.62i·27-s − 10.2·29-s − 3.56·31-s + ⋯
L(s)  = 1  + 1.48i·3-s + (0.498 − 0.866i)5-s − 0.238i·7-s − 1.21·9-s + 1.85·11-s + 0.277i·13-s + (1.28 + 0.741i)15-s + 1.02i·17-s + 1.21·19-s + 0.355·21-s + 0.489i·23-s + (−0.502 − 0.864i)25-s − 0.313i·27-s − 1.90·29-s − 0.639·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.498 - 0.866i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.498 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45806 + 0.843332i\)
\(L(\frac12)\) \(\approx\) \(1.45806 + 0.843332i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 1.93i)T \)
13 \( 1 - iT \)
good3 \( 1 - 2.57iT - 3T^{2} \)
7 \( 1 + 0.632iT - 7T^{2} \)
11 \( 1 - 6.16T + 11T^{2} \)
17 \( 1 - 4.23iT - 17T^{2} \)
19 \( 1 - 5.31T + 19T^{2} \)
23 \( 1 - 2.34iT - 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 + 3.56T + 31T^{2} \)
37 \( 1 + 7.09iT - 37T^{2} \)
41 \( 1 - 5.01T + 41T^{2} \)
43 \( 1 - 8.32iT - 43T^{2} \)
47 \( 1 + 3.36iT - 47T^{2} \)
53 \( 1 + 2.60iT - 53T^{2} \)
59 \( 1 + 4.29T + 59T^{2} \)
61 \( 1 - 1.37T + 61T^{2} \)
67 \( 1 - 8.36iT - 67T^{2} \)
71 \( 1 + 0.481T + 71T^{2} \)
73 \( 1 + 7.11iT - 73T^{2} \)
79 \( 1 - 8.90T + 79T^{2} \)
83 \( 1 + 15.5iT - 83T^{2} \)
89 \( 1 - 1.29T + 89T^{2} \)
97 \( 1 - 1.88iT - 97T^{2} \)
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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

−10.96411047958319991141498123361, −9.876688726195000741651201235072, −9.264172356091027952547622442328, −8.930384332527732295391650221336, −7.49272123762478467511526953746, −6.08932294129504140303198079171, −5.30800279797552182305157588772, −4.14026405415693818889729564048, −3.69669896238922479575889006065, −1.54633977486016815100548592573, 1.24771495739901540511014618257, 2.40666007709951260300290622231, 3.62926577332888476764766584018, 5.48635199280039378217554724111, 6.36467704176903810744575455782, 7.06570825384809408779161413056, 7.65219850221174053173755208937, 9.044121509813531970925818046968, 9.625547695096219644566806052459, 11.02013060942463605139016583023

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