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Properties

Label 2-1520-1520.949-c0-0-1
Degree $2$
Conductor $1520$
Sign $-0.923 + 0.382i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.555 + 0.831i)2-s + (−1.17 + 1.17i)3-s + (−0.382 − 0.923i)4-s + (−0.707 − 0.707i)5-s + (−0.324 − 1.63i)6-s + (0.980 + 0.195i)8-s − 1.76i·9-s + (0.980 − 0.195i)10-s + (1.30 + 1.30i)11-s + (1.53 + 0.636i)12-s + (−0.275 + 0.275i)13-s + 1.66·15-s + (−0.707 + 0.707i)16-s + (1.46 + 0.980i)18-s + (−0.707 + 0.707i)19-s + (−0.382 + 0.923i)20-s + ⋯
L(s)  = 1  + (−0.555 + 0.831i)2-s + (−1.17 + 1.17i)3-s + (−0.382 − 0.923i)4-s + (−0.707 − 0.707i)5-s + (−0.324 − 1.63i)6-s + (0.980 + 0.195i)8-s − 1.76i·9-s + (0.980 − 0.195i)10-s + (1.30 + 1.30i)11-s + (1.53 + 0.636i)12-s + (−0.275 + 0.275i)13-s + 1.66·15-s + (−0.707 + 0.707i)16-s + (1.46 + 0.980i)18-s + (−0.707 + 0.707i)19-s + (−0.382 + 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2892393233\)
\(L(\frac12)\) \(\approx\) \(0.2892393233\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.555 - 0.831i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (1.17 - 1.17i)T - iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
13 \( 1 + (0.275 - 0.275i)T - iT^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.785 + 0.785i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1.17 - 1.17i)T + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
67 \( 1 + (1.38 - 1.38i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 0.390T + T^{2} \)
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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

−10.03767183523033742994563738069, −9.226623600479366340293032885811, −8.821464913615197838407090860286, −7.59878287064722081077771594094, −6.84774020165620040048242871697, −5.98853915650004094077214970592, −5.17428139143533491102212751399, −4.27666700409041233696369641087, −4.11056089117923831645586868083, −1.46380142472081984731558552198, 0.35728694257327458602788855850, 1.59002130879454004207461233310, 2.91223227976647815283824858928, 3.87088620126815219826979013832, 5.04304205469594750249443932759, 6.38558682967454929431749858066, 6.69018265522571171622199419546, 7.64279872849416817743612998423, 8.322641830896030274086596205560, 9.171494899428220476725189320054

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