Properties

Label 2-520-5.4-c1-0-2
Degree $2$
Conductor $520$
Sign $-0.981 + 0.193i$
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  + 3.05i·3-s + (−2.19 + 0.433i)5-s + 3.34i·7-s − 6.34·9-s + 4.88·11-s i·13-s + (−1.32 − 6.70i)15-s + 2.38i·17-s − 7.22·19-s − 10.2·21-s − 3.32i·23-s + (4.62 − 1.90i)25-s − 10.2i·27-s − 3.95·29-s + 2.96·31-s + ⋯
L(s)  = 1  + 1.76i·3-s + (−0.981 + 0.193i)5-s + 1.26i·7-s − 2.11·9-s + 1.47·11-s − 0.277i·13-s + (−0.342 − 1.73i)15-s + 0.578i·17-s − 1.65·19-s − 2.23·21-s − 0.694i·23-s + (0.924 − 0.380i)25-s − 1.97i·27-s − 0.733·29-s + 0.532·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.981 + 0.193i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.981 + 0.193i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0910540 - 0.929923i\)
\(L(\frac12)\) \(\approx\) \(0.0910540 - 0.929923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.19 - 0.433i)T \)
13 \( 1 + iT \)
good3 \( 1 - 3.05iT - 3T^{2} \)
7 \( 1 - 3.34iT - 7T^{2} \)
11 \( 1 - 4.88T + 11T^{2} \)
17 \( 1 - 2.38iT - 17T^{2} \)
19 \( 1 + 7.22T + 19T^{2} \)
23 \( 1 + 3.32iT - 23T^{2} \)
29 \( 1 + 3.95T + 29T^{2} \)
31 \( 1 - 2.96T + 31T^{2} \)
37 \( 1 + 3.42iT - 37T^{2} \)
41 \( 1 - 4.43T + 41T^{2} \)
43 \( 1 - 3.32iT - 43T^{2} \)
47 \( 1 - 0.651iT - 47T^{2} \)
53 \( 1 - 7.84iT - 53T^{2} \)
59 \( 1 - 7.66T + 59T^{2} \)
61 \( 1 + 15.0T + 61T^{2} \)
67 \( 1 - 10.4iT - 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 + 8.38T + 79T^{2} \)
83 \( 1 - 14.6iT - 83T^{2} \)
89 \( 1 - 8.50T + 89T^{2} \)
97 \( 1 - 14.8iT - 97T^{2} \)
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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

−11.14888432878396704891729010871, −10.58815268353645903072640758591, −9.453816374455354863927156524628, −8.819916559513815815003173383882, −8.235375675318612095140611801986, −6.53948923729614017472726064367, −5.64916283581427305294155441238, −4.36506273722018240368311626709, −3.92618849234627973025502420122, −2.67601269624644218940535984285, 0.56412106327779268722117982568, 1.74056703651312963783787022176, 3.50326100497040140227899327500, 4.49325556815583383037018306119, 6.24641551291494118973250347971, 6.91777601405467365846678486858, 7.50683723339655611625373073527, 8.334380493708075028255342320438, 9.232165159349422499950298335272, 10.76216442805426051177924704544

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