Properties

Label 2-1520-5.4-c1-0-27
Degree $2$
Conductor $1520$
Sign $1$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
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  − 0.874i·3-s + 2.23·5-s + 2.82i·7-s + 2.23·9-s + 0.763·11-s − 5.45i·13-s − 1.95i·15-s + 7.40i·17-s − 19-s + 2.47·21-s + 1.08i·23-s + 5.00·25-s − 4.57i·27-s + 4.47·29-s + 4·31-s + ⋯
L(s)  = 1  − 0.504i·3-s + 0.999·5-s + 1.06i·7-s + 0.745·9-s + 0.230·11-s − 1.51i·13-s − 0.504i·15-s + 1.79i·17-s − 0.229·19-s + 0.539·21-s + 0.225i·23-s + 1.00·25-s − 0.880i·27-s + 0.830·29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.256127227\)
\(L(\frac12)\) \(\approx\) \(2.256127227\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T \)
19 \( 1 + T \)
good3 \( 1 + 0.874iT - 3T^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 - 0.763T + 11T^{2} \)
13 \( 1 + 5.45iT - 13T^{2} \)
17 \( 1 - 7.40iT - 17T^{2} \)
23 \( 1 - 1.08iT - 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2.62iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 - 8.48iT - 47T^{2} \)
53 \( 1 - 2.62iT - 53T^{2} \)
59 \( 1 + 1.52T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 - 5.24iT - 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + 13.7iT - 83T^{2} \)
89 \( 1 + 2.94T + 89T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−9.505311258893448360383406267947, −8.528384915927602925103755657517, −8.072531279329500846321863866102, −6.89199686909356662442319023055, −6.08678966076541757880115864477, −5.63888996969578368688672453369, −4.55380762098603427637189860559, −3.20096910062222297460504201692, −2.19174949254656807452003272147, −1.27329487305174095836253820266, 1.08853593004070106684044245866, 2.26374560378313040173124509091, 3.57236117439916879968614843310, 4.60092908589301462952784790211, 4.98178539175047273138265417363, 6.59405542835070585831169088581, 6.73338720964366953546835899141, 7.76494788730919408381344375565, 9.077632763134642413390577396688, 9.471766248116396315255224588331

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