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Properties

Label 2-152880-1.1-c1-0-178
Degree $2$
Conductor $152880$
Sign $-1$
Analytic cond. $1220.75$
Root an. cond. $34.9392$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s + 3·11-s − 13-s + 15-s − 3·17-s + 2·19-s − 9·23-s + 25-s + 27-s − 6·29-s − 4·31-s + 3·33-s − 37-s − 39-s + 3·41-s + 10·43-s + 45-s − 3·51-s + 6·53-s + 3·55-s + 2·57-s − 9·59-s − 2·61-s − 65-s + 67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.458·19-s − 1.87·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.522·33-s − 0.164·37-s − 0.160·39-s + 0.468·41-s + 1.52·43-s + 0.149·45-s − 0.420·51-s + 0.824·53-s + 0.404·55-s + 0.264·57-s − 1.17·59-s − 0.256·61-s − 0.124·65-s + 0.122·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152880\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1220.75\)
Root analytic conductor: \(34.9392\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 152880,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−13.65838621700240, −13.13100800710626, −12.65758483389253, −12.14854069085509, −11.73897887806187, −11.18069362268490, −10.59778150342574, −10.23176379143952, −9.489207776885925, −9.233460997275339, −9.002289142343262, −8.173458211409485, −7.740929793279827, −7.322192525954853, −6.674264874507207, −6.202383118147972, −5.714145964678901, −5.152857100692155, −4.371486493497822, −3.968261109304370, −3.540844937764530, −2.722184943791121, −2.107291891646831, −1.774020182561836, −0.9369441515870900, 0, 0.9369441515870900, 1.774020182561836, 2.107291891646831, 2.722184943791121, 3.540844937764530, 3.968261109304370, 4.371486493497822, 5.152857100692155, 5.714145964678901, 6.202383118147972, 6.674264874507207, 7.322192525954853, 7.740929793279827, 8.173458211409485, 9.002289142343262, 9.233460997275339, 9.489207776885925, 10.23176379143952, 10.59778150342574, 11.18069362268490, 11.73897887806187, 12.14854069085509, 12.65758483389253, 13.13100800710626, 13.65838621700240

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