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Properties

Label 2-520-104.101-c1-0-16
Degree $2$
Conductor $520$
Sign $0.993 + 0.118i$
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  + (0.237 − 1.39i)2-s + (−0.158 + 0.0913i)3-s + (−1.88 − 0.661i)4-s − 5-s + (0.0898 + 0.242i)6-s + (2.61 + 1.50i)7-s + (−1.36 + 2.47i)8-s + (−1.48 + 2.56i)9-s + (−0.237 + 1.39i)10-s + (1.37 + 2.38i)11-s + (0.358 − 0.0677i)12-s + (2.87 + 2.17i)13-s + (2.72 − 3.28i)14-s + (0.158 − 0.0913i)15-s + (3.12 + 2.49i)16-s + (1.19 − 2.06i)17-s + ⋯
L(s)  = 1  + (0.167 − 0.985i)2-s + (−0.0913 + 0.0527i)3-s + (−0.943 − 0.330i)4-s − 0.447·5-s + (0.0366 + 0.0988i)6-s + (0.987 + 0.570i)7-s + (−0.484 + 0.875i)8-s + (−0.494 + 0.856i)9-s + (−0.0749 + 0.440i)10-s + (0.415 + 0.720i)11-s + (0.103 − 0.0195i)12-s + (0.797 + 0.603i)13-s + (0.727 − 0.877i)14-s + (0.0408 − 0.0235i)15-s + (0.781 + 0.623i)16-s + (0.289 − 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.993 + 0.118i$
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: \((2,\ 520,\ (\ :1/2),\ 0.993 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31959 - 0.0781510i\)
\(L(\frac12)\) \(\approx\) \(1.31959 - 0.0781510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.237 + 1.39i)T \)
5 \( 1 + T \)
13 \( 1 + (-2.87 - 2.17i)T \)
good3 \( 1 + (0.158 - 0.0913i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.61 - 1.50i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.37 - 2.38i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.19 + 2.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.343 + 0.594i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.16 + 2.01i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.86 - 1.65i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.49iT - 31T^{2} \)
37 \( 1 + (-3.59 - 6.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.68 + 5.01i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.97 + 1.14i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.99iT - 47T^{2} \)
53 \( 1 - 6.03iT - 53T^{2} \)
59 \( 1 + (-0.982 + 1.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.4 - 7.21i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.739 + 1.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (13.5 + 7.84i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.24iT - 73T^{2} \)
79 \( 1 + 5.23T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + (-10.0 + 5.81i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.21 + 3.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.17662373314975107241282859716, −10.19157490418017897412217524859, −9.062464277522296923318190634739, −8.451826244022722488054280958337, −7.47350138410728617507771316830, −5.90457018776185774163657201330, −4.90562802271531819916375181964, −4.18616146067379012710960316355, −2.71750927550869708420116777200, −1.56059174048572509506572036328, 0.858764928759839820817049746159, 3.48608525380979404988167730970, 4.15301978484542968518559144335, 5.53957228331809691602870182419, 6.17737520136635939200968803590, 7.34858936186787290025511406428, 8.150023278615169965598520672247, 8.725770890127479443618458136566, 9.828408363611299247219310813864, 11.09564678919432019147933830769

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