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Properties

Label 2-520-104.101-c1-0-22
Degree $2$
Conductor $520$
Sign $-0.134 - 0.990i$
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  + (0.837 + 1.13i)2-s + (0.654 − 0.377i)3-s + (−0.596 + 1.90i)4-s + 5-s + (0.979 + 0.429i)6-s + (0.518 + 0.299i)7-s + (−2.67 + 0.919i)8-s + (−1.21 + 2.10i)9-s + (0.837 + 1.13i)10-s + (0.495 + 0.858i)11-s + (0.331 + 1.47i)12-s + (1.20 + 3.39i)13-s + (0.0933 + 0.842i)14-s + (0.654 − 0.377i)15-s + (−3.28 − 2.27i)16-s + (3.48 − 6.03i)17-s + ⋯
L(s)  = 1  + (0.592 + 0.805i)2-s + (0.377 − 0.218i)3-s + (−0.298 + 0.954i)4-s + 0.447·5-s + (0.399 + 0.175i)6-s + (0.196 + 0.113i)7-s + (−0.945 + 0.325i)8-s + (−0.404 + 0.701i)9-s + (0.264 + 0.360i)10-s + (0.149 + 0.258i)11-s + (0.0955 + 0.425i)12-s + (0.333 + 0.942i)13-s + (0.0249 + 0.225i)14-s + (0.169 − 0.0975i)15-s + (−0.822 − 0.569i)16-s + (0.844 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.134 - 0.990i$
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: \((2,\ 520,\ (\ :1/2),\ -0.134 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45270 + 1.66278i\)
\(L(\frac12)\) \(\approx\) \(1.45270 + 1.66278i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.837 - 1.13i)T \)
5 \( 1 - T \)
13 \( 1 + (-1.20 - 3.39i)T \)
good3 \( 1 + (-0.654 + 0.377i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.518 - 0.299i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.495 - 0.858i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.48 + 6.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.07 - 3.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.57 - 4.45i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.70 + 2.13i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.2iT - 31T^{2} \)
37 \( 1 + (-3.38 - 5.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.99 - 1.15i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.49 + 4.90i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.84iT - 47T^{2} \)
53 \( 1 - 1.02iT - 53T^{2} \)
59 \( 1 + (-0.213 + 0.369i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.52 - 4.34i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.26 + 9.12i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.90 - 1.67i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + 1.48T + 79T^{2} \)
83 \( 1 + 8.05T + 83T^{2} \)
89 \( 1 + (-4.06 + 2.34i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.6 - 7.29i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.54979825765130585681108300201, −9.994268493239625718075127883126, −9.143346112321845899136117940300, −8.224484013592221186814110495124, −7.48556525991064977572672579796, −6.53027187224930783525738117041, −5.51690882932725156534376011194, −4.68941891143661579606073574878, −3.36135918698594022279948546382, −2.11279209398078251222955923284, 1.16230718795732981486476656925, 2.79521532840701270198668978668, 3.57918571676848499655029044445, 4.79371183248212424991486254710, 5.85863557811674839332168235524, 6.61630774153944129744541248598, 8.376111293183044150612990934985, 8.891574601679425744358574564621, 10.04541025319371617172142447494, 10.59565419543725196186088904099

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