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Properties

Label 2-1520-19.11-c1-0-12
Degree $2$
Conductor $1520$
Sign $0.975 - 0.221i$
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  + (−1.69 + 2.93i)3-s + (0.5 − 0.866i)5-s − 2.29·7-s + (−4.23 − 7.32i)9-s − 5.21·11-s + (−0.116 − 0.201i)13-s + (1.69 + 2.93i)15-s + (−2.19 + 3.80i)17-s + (2.13 − 3.79i)19-s + (3.89 − 6.74i)21-s + (0.643 + 1.11i)23-s + (−0.499 − 0.866i)25-s + 18.4·27-s + (3.31 + 5.73i)29-s − 0.286·31-s + ⋯
L(s)  = 1  + (−0.977 + 1.69i)3-s + (0.223 − 0.387i)5-s − 0.869·7-s + (−1.41 − 2.44i)9-s − 1.57·11-s + (−0.0322 − 0.0558i)13-s + (0.437 + 0.757i)15-s + (−0.533 + 0.923i)17-s + (0.490 − 0.871i)19-s + (0.849 − 1.47i)21-s + (0.134 + 0.232i)23-s + (−0.0999 − 0.173i)25-s + 3.55·27-s + (0.614 + 1.06i)29-s − 0.0514·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.975 - 0.221i$
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: \((2,\ 1520,\ (\ :1/2),\ 0.975 - 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6231720015\)
\(L(\frac12)\) \(\approx\) \(0.6231720015\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.13 + 3.79i)T \)
good3 \( 1 + (1.69 - 2.93i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 2.29T + 7T^{2} \)
11 \( 1 + 5.21T + 11T^{2} \)
13 \( 1 + (0.116 + 0.201i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.19 - 3.80i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.643 - 1.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.31 - 5.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.286T + 31T^{2} \)
37 \( 1 - 9.07T + 37T^{2} \)
41 \( 1 + (2.14 - 3.72i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.53 + 9.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.57 - 4.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.54 + 6.14i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.24 + 5.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.47 + 7.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.09 - 8.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.36 + 12.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.282 - 0.489i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.18 - 3.78i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.67T + 83T^{2} \)
89 \( 1 + (2.21 + 3.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.75 + 3.04i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.609682543176762800134814735818, −9.066558220066379871114544321340, −8.124550502684744268838678919280, −6.74976071523612971832785006526, −5.97694548664367045066855644171, −5.21071835798154453122482083293, −4.68514141572333550939473122866, −3.62756144350191857213285133949, −2.76101611572006451356342222702, −0.40717934256632892597683633518, 0.77160947103113199468107845291, 2.32171944112830156384509171848, 2.85406982170016119719335575927, 4.73201090259461657509258533244, 5.71627892816777393873011637277, 6.15614592889118615969438475417, 7.01652235529517435587487226640, 7.62480732304010852388693242487, 8.241799092290988339073772678736, 9.580186536470709814454089537377

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