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Properties

Label 2-1520-19.7-c1-0-25
Degree $2$
Conductor $1520$
Sign $-0.0977 + 0.995i$
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  + (−1 − 1.73i)3-s + (0.5 + 0.866i)5-s + 4·7-s + (−0.499 + 0.866i)9-s − 3·11-s + (−1 + 1.73i)13-s + (0.999 − 1.73i)15-s + (−3 − 5.19i)17-s + (3.5 − 2.59i)19-s + (−4 − 6.92i)21-s + (−0.499 + 0.866i)25-s − 4.00·27-s + (1.5 − 2.59i)29-s + 7·31-s + (3 + 5.19i)33-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + (0.223 + 0.387i)5-s + 1.51·7-s + (−0.166 + 0.288i)9-s − 0.904·11-s + (−0.277 + 0.480i)13-s + (0.258 − 0.447i)15-s + (−0.727 − 1.26i)17-s + (0.802 − 0.596i)19-s + (−0.872 − 1.51i)21-s + (−0.0999 + 0.173i)25-s − 0.769·27-s + (0.278 − 0.482i)29-s + 1.25·31-s + (0.522 + 0.904i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.504775435\)
\(L(\frac12)\) \(\approx\) \(1.504775435\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-3.5 + 2.59i)T \)
good3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.5 + 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4 + 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.252508612673018219940622764878, −8.169917750721099319846166767947, −7.54303897522590284228855422604, −6.95021391048562427227319346264, −6.07142063306582416408749670308, −5.08572546577789541153812699227, −4.53633511808271365685234792374, −2.77423181198520951646520141157, −1.94199116530934181466254624923, −0.69533860154188572812545058410, 1.33396587002979304530045835180, 2.60809874318430625096960147622, 4.15809240524133906136690797769, 4.67577895017360608919188423230, 5.41550302269319945327835617650, 6.02518632809136598173198079807, 7.57760972250868332355382845065, 8.061263372806999086641623104913, 8.882328767680833074056068308737, 9.899202508924128540373157227463

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