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Properties

Label 2-520-104.69-c1-0-24
Degree $2$
Conductor $520$
Sign $0.974 + 0.224i$
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  + (−0.0430 − 1.41i)2-s + (2.59 + 1.49i)3-s + (−1.99 + 0.121i)4-s + 5-s + (2.00 − 3.73i)6-s + (−0.668 + 0.385i)7-s + (0.257 + 2.81i)8-s + (2.98 + 5.16i)9-s + (−0.0430 − 1.41i)10-s + (−0.183 + 0.317i)11-s + (−5.35 − 2.67i)12-s + (2.71 − 2.36i)13-s + (0.574 + 0.928i)14-s + (2.59 + 1.49i)15-s + (3.97 − 0.485i)16-s + (1.37 + 2.37i)17-s + ⋯
L(s)  = 1  + (−0.0304 − 0.999i)2-s + (1.49 + 0.864i)3-s + (−0.998 + 0.0607i)4-s + 0.447·5-s + (0.818 − 1.52i)6-s + (−0.252 + 0.145i)7-s + (0.0911 + 0.995i)8-s + (0.994 + 1.72i)9-s + (−0.0135 − 0.447i)10-s + (−0.0552 + 0.0956i)11-s + (−1.54 − 0.771i)12-s + (0.753 − 0.657i)13-s + (0.153 + 0.248i)14-s + (0.669 + 0.386i)15-s + (0.992 − 0.121i)16-s + (0.332 + 0.575i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.974 + 0.224i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.974 + 0.224i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19933 - 0.249899i\)
\(L(\frac12)\) \(\approx\) \(2.19933 - 0.249899i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0430 + 1.41i)T \)
5 \( 1 - T \)
13 \( 1 + (-2.71 + 2.36i)T \)
good3 \( 1 + (-2.59 - 1.49i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.668 - 0.385i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.183 - 0.317i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.37 - 2.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.199 - 0.345i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.390 - 0.676i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.82 + 1.05i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.83iT - 31T^{2} \)
37 \( 1 + (4.55 - 7.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.57 - 3.79i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.38 - 3.10i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.57iT - 47T^{2} \)
53 \( 1 + 4.85iT - 53T^{2} \)
59 \( 1 + (6.27 + 10.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.23 - 0.712i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.10 - 3.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.22 + 0.707i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 16.3iT - 73T^{2} \)
79 \( 1 + 4.89T + 79T^{2} \)
83 \( 1 + 2.70T + 83T^{2} \)
89 \( 1 + (10.0 + 5.82i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.2 + 6.48i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.52645306273854197021425214715, −9.878560308416904263005613558514, −9.334912849098707373355013030762, −8.406805295495744450394235533397, −7.86370717026659879194503477720, −5.95518785724294527360351597503, −4.72604376305583212123962736152, −3.68004631478244396131897317531, −2.98147159495619936159925052319, −1.82115802237513887872885982519, 1.42791936693660863274840154192, 3.00628778133818768570685640997, 4.06074010338716614691191614362, 5.56585829426747353214057327105, 6.72682475722575212115911754995, 7.21853659583952590755064474213, 8.210309570014313917684543542686, 8.966431446236856839968579909888, 9.425750353122368022044346281267, 10.60281271894754530288946509840

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