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Properties

Label 2-520-65.64-c1-0-13
Degree $2$
Conductor $520$
Sign $0.761 + 0.647i$
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  − 0.773i·3-s + (1.98 − 1.02i)5-s + 1.12·7-s + 2.40·9-s + 2.52i·11-s + (−1.37 − 3.33i)13-s + (−0.789 − 1.53i)15-s − 0.870i·17-s + 6.27i·19-s − 0.870i·21-s − 2.81i·23-s + (2.91 − 4.06i)25-s − 4.17i·27-s − 0.176·29-s − 0.388i·31-s + ⋯
L(s)  = 1  − 0.446i·3-s + (0.889 − 0.456i)5-s + 0.425·7-s + 0.800·9-s + 0.760i·11-s + (−0.381 − 0.924i)13-s + (−0.203 − 0.397i)15-s − 0.211i·17-s + 1.43i·19-s − 0.189i·21-s − 0.586i·23-s + (0.583 − 0.812i)25-s − 0.803i·27-s − 0.0328·29-s − 0.0698i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.761 + 0.647i$
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: \((2,\ 520,\ (\ :1/2),\ 0.761 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69277 - 0.622608i\)
\(L(\frac12)\) \(\approx\) \(1.69277 - 0.622608i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.98 + 1.02i)T \)
13 \( 1 + (1.37 + 3.33i)T \)
good3 \( 1 + 0.773iT - 3T^{2} \)
7 \( 1 - 1.12T + 7T^{2} \)
11 \( 1 - 2.52iT - 11T^{2} \)
17 \( 1 + 0.870iT - 17T^{2} \)
19 \( 1 - 6.27iT - 19T^{2} \)
23 \( 1 + 2.81iT - 23T^{2} \)
29 \( 1 + 0.176T + 29T^{2} \)
31 \( 1 + 0.388iT - 31T^{2} \)
37 \( 1 - 3.12T + 37T^{2} \)
41 \( 1 + 7.25iT - 41T^{2} \)
43 \( 1 + 4.52iT - 43T^{2} \)
47 \( 1 + 1.12T + 47T^{2} \)
53 \( 1 - 9.83iT - 53T^{2} \)
59 \( 1 - 12.1iT - 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 + 4.37iT - 71T^{2} \)
73 \( 1 + 4.70T + 73T^{2} \)
79 \( 1 + 8.31T + 79T^{2} \)
83 \( 1 + 9.15T + 83T^{2} \)
89 \( 1 - 6.15iT - 89T^{2} \)
97 \( 1 - 1.55T + 97T^{2} \)
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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

−10.37715043579732770046743051276, −10.11432955624298570368964408280, −9.034724274563823483094871465044, −7.976587831811759546212910733595, −7.23699199403839414579213658976, −6.11020233211826157792150330055, −5.19531851779143253139319062839, −4.17772850600034976520188607975, −2.39996787610079345999889883147, −1.32088740219816192054630510980, 1.65363070470037908090416834079, 3.01282771885362301650031284521, 4.38091122251793873821007132733, 5.26520391729427714890506664225, 6.45668881895852296288889660693, 7.17155456288815571989411243659, 8.441214950151437721211002035718, 9.484137513237650571285687312137, 9.895944431486315547510046979384, 11.09203994583435442109110931748

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