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Properties

Label 2-1520-19.11-c1-0-6
Degree $2$
Conductor $1520$
Sign $-0.813 - 0.582i$
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  + (−1.25 + 2.17i)3-s + (0.5 − 0.866i)5-s − 3.50·7-s + (−1.64 − 2.84i)9-s + 4.50·11-s + (2.5 + 4.33i)13-s + (1.25 + 2.17i)15-s + (−0.0793 + 0.137i)17-s + (4.26 − 0.920i)19-s + (4.39 − 7.61i)21-s + (0.579 + 1.00i)23-s + (−0.499 − 0.866i)25-s + 0.714·27-s + (1.75 + 3.03i)29-s + 2.28·31-s + ⋯
L(s)  = 1  + (−0.723 + 1.25i)3-s + (0.223 − 0.387i)5-s − 1.32·7-s + (−0.547 − 0.948i)9-s + 1.35·11-s + (0.693 + 1.20i)13-s + (0.323 + 0.560i)15-s + (−0.0192 + 0.0333i)17-s + (0.977 − 0.211i)19-s + (0.959 − 1.66i)21-s + (0.120 + 0.209i)23-s + (−0.0999 − 0.173i)25-s + 0.137·27-s + (0.325 + 0.563i)29-s + 0.410·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.813 - 0.582i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ -0.813 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9804481828\)
\(L(\frac12)\) \(\approx\) \(0.9804481828\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.26 + 0.920i)T \)
good3 \( 1 + (1.25 - 2.17i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 3.50T + 7T^{2} \)
11 \( 1 - 4.50T + 11T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.0793 - 0.137i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.579 - 1.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.75 - 3.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.28T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + (3.03 - 5.26i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.67 - 2.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.53 - 2.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.87 - 4.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.53 + 2.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.436 - 0.756i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.22 + 7.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.11 - 14.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.57 + 6.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.06 - 8.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.85T + 83T^{2} \)
89 \( 1 + (-0.556 - 0.963i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.809 - 1.40i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.682588872240949443337672855136, −9.260006964952486706698050782273, −8.654102068815805166671344738586, −7.00354351708473399298842234315, −6.44002014884025505809570914510, −5.67705162834415201386903862316, −4.70997868820301282381402693261, −3.92378757235848158182085326267, −3.23177797785476918331971850757, −1.34408463933441205796984292032, 0.47292153138323557865301189351, 1.57928678146129614105410370352, 3.01256951842232508180883009105, 3.77475849069243258570198178211, 5.47188080743017124180749515629, 6.02395282628553196293622365638, 6.77972013175933117395599969826, 7.11914361412449133504776725376, 8.248735832146240330498200948559, 9.151537348616897590367693753105

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