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Properties

Label 2-1520-19.11-c1-0-14
Degree $2$
Conductor $1520$
Sign $0.910 - 0.412i$
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  + (0.610 − 1.05i)3-s + (0.5 − 0.866i)5-s + 0.221·7-s + (0.753 + 1.30i)9-s + 0.778·11-s + (2.5 + 4.33i)13-s + (−0.610 − 1.05i)15-s + (−3.53 + 6.12i)17-s + (−1.33 + 4.15i)19-s + (0.135 − 0.234i)21-s + (4.03 + 6.99i)23-s + (−0.499 − 0.866i)25-s + 5.50·27-s + (−0.110 − 0.192i)29-s − 2.50·31-s + ⋯
L(s)  = 1  + (0.352 − 0.610i)3-s + (0.223 − 0.387i)5-s + 0.0838·7-s + (0.251 + 0.435i)9-s + 0.234·11-s + (0.693 + 1.20i)13-s + (−0.157 − 0.273i)15-s + (−0.858 + 1.48i)17-s + (−0.305 + 0.952i)19-s + (0.0295 − 0.0512i)21-s + (0.842 + 1.45i)23-s + (−0.0999 − 0.173i)25-s + 1.05·27-s + (−0.0206 − 0.0356i)29-s − 0.450·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.910 - 0.412i$
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: \((2,\ 1520,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.019192303\)
\(L(\frac12)\) \(\approx\) \(2.019192303\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 - 4.15i)T \)
good3 \( 1 + (-0.610 + 1.05i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 0.221T + 7T^{2} \)
11 \( 1 - 0.778T + 11T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.53 - 6.12i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.03 - 6.99i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.110 + 0.192i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.50T + 31T^{2} \)
37 \( 1 + 1.90T + 37T^{2} \)
41 \( 1 + (-3.61 + 6.26i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.64 + 6.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.39 + 2.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.19 + 3.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.39 - 2.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.29 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.28 + 9.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.92 - 8.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.03 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.792 + 1.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.52T + 83T^{2} \)
89 \( 1 + (1.57 + 2.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.18 + 5.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.240324474037313296745977415134, −8.743766624621126640636664031547, −7.978537595078889150040080391854, −7.12218241522382325695604994519, −6.36641076899067438404485163383, −5.50151509610585085238729289854, −4.35521836110666599235972381823, −3.61282171090185466277545433525, −1.96800834543383774011744135318, −1.54883857646813537268135246014, 0.803641513795898249676624971418, 2.60937356737343603659511023245, 3.22361899889921561813444484197, 4.40454429756971503093498159376, 5.05805934496510621177520231966, 6.33860259763557541747952291793, 6.84049220560093037563548274499, 7.902649260838125456677668781731, 8.861617800837242837735967291091, 9.320906456611936700878542002108

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