Properties

Label 2-1520-5.4-c1-0-10
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  + 2.28i·3-s − 2.23·5-s + 2.82i·7-s − 2.23·9-s + 5.23·11-s + 4.03i·13-s − 5.11i·15-s + 1.08i·17-s − 19-s − 6.47·21-s + 7.40i·23-s + 5.00·25-s + 1.74i·27-s − 4.47·29-s + 4·31-s + ⋯
L(s)  = 1  + 1.32i·3-s − 0.999·5-s + 1.06i·7-s − 0.745·9-s + 1.57·11-s + 1.11i·13-s − 1.32i·15-s + 0.262i·17-s − 0.229·19-s − 1.41·21-s + 1.54i·23-s + 1.00·25-s + 0.336i·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

\(L(1)\) \(\approx\) \(1.244546928\)
\(L(\frac12)\) \(\approx\) \(1.244546928\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.28iT - 3T^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 - 5.23T + 11T^{2} \)
13 \( 1 - 4.03iT - 13T^{2} \)
17 \( 1 - 1.08iT - 17T^{2} \)
23 \( 1 - 7.40iT - 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 6.86iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 - 8.48iT - 47T^{2} \)
53 \( 1 + 6.86iT - 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 1.70T + 61T^{2} \)
67 \( 1 - 1.62iT - 67T^{2} \)
71 \( 1 - 1.52T + 71T^{2} \)
73 \( 1 + 13.7iT - 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 5.24iT - 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 4.44iT - 97T^{2} \)
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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

−9.569351058530270919248124202637, −9.142311926297458592004019841102, −8.664152288645438628781776669386, −7.49506399577974839075843619224, −6.58398322769346663071247319122, −5.62069275659300010840208802164, −4.64367850574787660840022587846, −3.93149059235761268457382891037, −3.37792996068683155764169322055, −1.77219049009842438974043788465, 0.54542025728583361434968334480, 1.35922589834201333082967658028, 2.91104047446450596028450771723, 3.92299089664035399741985989539, 4.69478788221482684592070565224, 6.21733726424002316977766055676, 6.78600600103757604026136685597, 7.39393990544659752977321113716, 8.112101416928282065364709412733, 8.713406643219459394982778866512

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