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Properties

Label 2-520-520.59-c1-0-2
Degree $2$
Conductor $520$
Sign $-0.928 - 0.372i$
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  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (−1.58 − 1.58i)5-s + (0.206 + 0.0552i)7-s + (−1.99 + 2i)8-s + (−1.5 − 2.59i)9-s + (2.73 + 1.58i)10-s + (0.0763 + 0.284i)11-s + (−2.89 + 2.15i)13-s − 0.301·14-s + (1.99 − 3.46i)16-s + (3 + 3i)18-s + (−1.98 + 7.40i)19-s + (−4.31 − 1.15i)20-s + (−0.208 − 0.361i)22-s + (−2.45 − 1.41i)23-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s + (−0.707 − 0.707i)5-s + (0.0779 + 0.0208i)7-s + (−0.707 + 0.707i)8-s + (−0.5 − 0.866i)9-s + (0.866 + 0.499i)10-s + (0.0230 + 0.0858i)11-s + (−0.802 + 0.597i)13-s − 0.0806·14-s + (0.499 − 0.866i)16-s + (0.707 + 0.707i)18-s + (−0.454 + 1.69i)19-s + (−0.965 − 0.258i)20-s + (−0.0444 − 0.0770i)22-s + (−0.512 − 0.295i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.928 - 0.372i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.928 - 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0154797 + 0.0801925i\)
\(L(\frac12)\) \(\approx\) \(0.0154797 + 0.0801925i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
5 \( 1 + (1.58 + 1.58i)T \)
13 \( 1 + (2.89 - 2.15i)T \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.206 - 0.0552i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.0763 - 0.284i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.98 - 7.40i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.45 + 1.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 31iT^{2} \)
37 \( 1 + (-2.46 - 9.20i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (10.0 - 2.68i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.21 + 7.21i)T - 47iT^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + (13.8 + 3.71i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (-2.22 - 8.30i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (84.0 + 48.5i)T^{2} \)
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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

−11.29778664353465091425788082772, −10.09863140232537958362176690719, −9.447708756039145538186687968175, −8.460279179229352025832044603646, −7.986778269310303686098594626397, −6.84360170331157070914456816458, −5.96034339829475822387152917335, −4.72478976457555463067129330433, −3.37168808093279478381474853519, −1.65842187587761281787419967416, 0.06253247766063439185884349371, 2.31181547553733617027535508295, 3.17410692755358204126425806304, 4.67155803543955906233649640761, 6.11118892813696934831997619861, 7.25782977984669646804184940117, 7.75193603852997895020963592079, 8.658379672682077323304214662918, 9.632804313767552646045416329347, 10.76304453727083783637548070841

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