Properties

Label 2-1520-1.1-c1-0-29
Degree $2$
Conductor $1520$
Sign $-1$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.470·3-s − 5-s − 2.71·7-s − 2.77·9-s + 5.55·11-s + 2.02·13-s − 0.470·15-s − 3.77·17-s − 19-s − 1.28·21-s + 5.77·23-s + 25-s − 2.71·27-s − 5.66·29-s − 7.55·31-s + 2.61·33-s + 2.71·35-s − 3.75·37-s + 0.954·39-s − 12.6·41-s + 9.43·43-s + 2.77·45-s − 11.1·47-s + 0.397·49-s − 1.77·51-s − 8.85·53-s − 5.55·55-s + ⋯
L(s)  = 1  + 0.271·3-s − 0.447·5-s − 1.02·7-s − 0.926·9-s + 1.67·11-s + 0.562·13-s − 0.121·15-s − 0.916·17-s − 0.229·19-s − 0.279·21-s + 1.20·23-s + 0.200·25-s − 0.523·27-s − 1.05·29-s − 1.35·31-s + 0.455·33-s + 0.459·35-s − 0.616·37-s + 0.152·39-s − 1.97·41-s + 1.43·43-s + 0.414·45-s − 1.62·47-s + 0.0567·49-s − 0.249·51-s − 1.21·53-s − 0.749·55-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
19 \( 1 + T \)
good3 \( 1 - 0.470T + 3T^{2} \)
7 \( 1 + 2.71T + 7T^{2} \)
11 \( 1 - 5.55T + 11T^{2} \)
13 \( 1 - 2.02T + 13T^{2} \)
17 \( 1 + 3.77T + 17T^{2} \)
23 \( 1 - 5.77T + 23T^{2} \)
29 \( 1 + 5.66T + 29T^{2} \)
31 \( 1 + 7.55T + 31T^{2} \)
37 \( 1 + 3.75T + 37T^{2} \)
41 \( 1 + 12.6T + 41T^{2} \)
43 \( 1 - 9.43T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + 8.85T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 10.6T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 9.45T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 - 4.94T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−9.077111960519254416733974828246, −8.540145357664334987504622043863, −7.35935278243352838393312261647, −6.57726309160933334728241837472, −6.03341621095946184504670611890, −4.78464883701679355126920616205, −3.61041505369148414840700250199, −3.25534928703236262790507389285, −1.72229109001735573255998072950, 0, 1.72229109001735573255998072950, 3.25534928703236262790507389285, 3.61041505369148414840700250199, 4.78464883701679355126920616205, 6.03341621095946184504670611890, 6.57726309160933334728241837472, 7.35935278243352838393312261647, 8.540145357664334987504622043863, 9.077111960519254416733974828246

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