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Properties

Label 2-520-104.101-c1-0-15
Degree $2$
Conductor $520$
Sign $-0.759 - 0.650i$
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  + (−0.683 + 1.23i)2-s + (−2.80 + 1.62i)3-s + (−1.06 − 1.69i)4-s − 5-s + (−0.0886 − 4.58i)6-s + (3.76 + 2.17i)7-s + (2.82 − 0.163i)8-s + (3.75 − 6.50i)9-s + (0.683 − 1.23i)10-s + (1.73 + 3.01i)11-s + (5.73 + 3.02i)12-s + (3.51 + 0.821i)13-s + (−5.26 + 3.17i)14-s + (2.80 − 1.62i)15-s + (−1.72 + 3.60i)16-s + (−0.0871 + 0.150i)17-s + ⋯
L(s)  = 1  + (−0.483 + 0.875i)2-s + (−1.62 + 0.936i)3-s + (−0.533 − 0.846i)4-s − 0.447·5-s + (−0.0361 − 1.87i)6-s + (1.42 + 0.822i)7-s + (0.998 − 0.0579i)8-s + (1.25 − 2.16i)9-s + (0.216 − 0.391i)10-s + (0.524 + 0.908i)11-s + (1.65 + 0.872i)12-s + (0.973 + 0.227i)13-s + (−1.40 + 0.849i)14-s + (0.725 − 0.418i)15-s + (−0.431 + 0.902i)16-s + (−0.0211 + 0.0366i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.759 - 0.650i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.759 - 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.250899 + 0.678562i\)
\(L(\frac12)\) \(\approx\) \(0.250899 + 0.678562i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.683 - 1.23i)T \)
5 \( 1 + T \)
13 \( 1 + (-3.51 - 0.821i)T \)
good3 \( 1 + (2.80 - 1.62i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.76 - 2.17i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.73 - 3.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.0871 - 0.150i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.52 + 6.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.971 - 1.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.50 + 1.44i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.00iT - 31T^{2} \)
37 \( 1 + (3.07 + 5.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.76 - 2.17i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.49 - 4.32i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.80iT - 47T^{2} \)
53 \( 1 - 6.77iT - 53T^{2} \)
59 \( 1 + (0.345 - 0.598i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.53 + 2.04i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.84 - 6.65i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.70 - 3.29i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.78iT - 73T^{2} \)
79 \( 1 + 4.84T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + (7.02 - 4.05i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.51 - 2.60i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.27172413891996994829742709864, −10.41322322623461078112313540408, −9.330232856766680740065864753232, −8.742989799306552269371930419989, −7.42934935952395695739629937996, −6.55284887351689353780499205280, −5.52844139116810097895252530323, −4.88896513982967880740899364881, −4.19378364810605714050278487839, −1.23041464378447211404285504359, 0.843136012976828336379892319886, 1.55413194417071643600485904752, 3.74975773846348963265573064791, 4.79927490862892479415182460376, 5.84437711928427085088532813799, 7.06190511452220821569090037581, 7.896595382970192789991792768019, 8.462472877835546868351544281465, 10.20189549406142803925910604144, 10.89403145270831062294725999547

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