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Properties

Label 2-520-40.29-c1-0-32
Degree $2$
Conductor $520$
Sign $0.997 - 0.0763i$
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  + (−0.641 + 1.26i)2-s − 1.27·3-s + (−1.17 − 1.61i)4-s + (2.22 + 0.182i)5-s + (0.815 − 1.60i)6-s + 0.963i·7-s + (2.79 − 0.446i)8-s − 1.38·9-s + (−1.66 + 2.69i)10-s − 4.73i·11-s + (1.49 + 2.05i)12-s + 13-s + (−1.21 − 0.618i)14-s + (−2.83 − 0.232i)15-s + (−1.22 + 3.80i)16-s + 1.12i·17-s + ⋯
L(s)  = 1  + (−0.453 + 0.891i)2-s − 0.733·3-s + (−0.588 − 0.808i)4-s + (0.996 + 0.0818i)5-s + (0.332 − 0.653i)6-s + 0.364i·7-s + (0.987 − 0.157i)8-s − 0.461·9-s + (−0.525 + 0.851i)10-s − 1.42i·11-s + (0.431 + 0.593i)12-s + 0.277·13-s + (−0.324 − 0.165i)14-s + (−0.731 − 0.0600i)15-s + (−0.307 + 0.951i)16-s + 0.272i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.997 - 0.0763i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.997 - 0.0763i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.906717 + 0.0346753i\)
\(L(\frac12)\) \(\approx\) \(0.906717 + 0.0346753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.641 - 1.26i)T \)
5 \( 1 + (-2.22 - 0.182i)T \)
13 \( 1 - T \)
good3 \( 1 + 1.27T + 3T^{2} \)
7 \( 1 - 0.963iT - 7T^{2} \)
11 \( 1 + 4.73iT - 11T^{2} \)
17 \( 1 - 1.12iT - 17T^{2} \)
19 \( 1 + 7.50iT - 19T^{2} \)
23 \( 1 + 3.18iT - 23T^{2} \)
29 \( 1 - 2.37iT - 29T^{2} \)
31 \( 1 - 6.64T + 31T^{2} \)
37 \( 1 - 2.66T + 37T^{2} \)
41 \( 1 - 4.61T + 41T^{2} \)
43 \( 1 + 3.36T + 43T^{2} \)
47 \( 1 + 0.590iT - 47T^{2} \)
53 \( 1 - 6.13T + 53T^{2} \)
59 \( 1 - 3.58iT - 59T^{2} \)
61 \( 1 + 8.35iT - 61T^{2} \)
67 \( 1 + 8.67T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 3.80iT - 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 2.23T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + 17.3iT - 97T^{2} \)
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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

−10.86926145337191768720560787936, −9.937185603055610331054902333196, −8.787087101798643919405116130903, −8.552627459361679811511471099725, −6.95444774172392037269482134079, −6.14624983991604470562721054133, −5.67207402353407606766006856958, −4.72830377762831708740591154555, −2.75204160953814340911837289532, −0.78517976926296219222225133670, 1.32526628204473385397972512590, 2.55474738060978988714526721804, 4.08956187007730534629999008670, 5.15922701537128500740306628590, 6.15174715189891329294747339195, 7.33015246719129904057297106023, 8.365919180132073632513766185475, 9.481486959406768849376198344173, 10.08449490357058045471924142054, 10.67669812200750292425695261929

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