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Properties

Label 2-135240-1.1-c1-0-39
Degree $2$
Conductor $135240$
Sign $1$
Analytic cond. $1079.89$
Root an. cond. $32.8617$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s + 6·11-s + 15-s + 17-s − 8·19-s + 23-s + 25-s + 27-s + 29-s + 5·31-s + 6·33-s + 5·37-s − 6·41-s − 5·43-s + 45-s + 8·47-s + 51-s + 7·53-s + 6·55-s − 8·57-s + 9·59-s − 10·61-s − 8·67-s + 69-s + 9·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.80·11-s + 0.258·15-s + 0.242·17-s − 1.83·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.185·29-s + 0.898·31-s + 1.04·33-s + 0.821·37-s − 0.937·41-s − 0.762·43-s + 0.149·45-s + 1.16·47-s + 0.140·51-s + 0.961·53-s + 0.809·55-s − 1.05·57-s + 1.17·59-s − 1.28·61-s − 0.977·67-s + 0.120·69-s + 1.06·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135240\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1079.89\)
Root analytic conductor: \(32.8617\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 135240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.530391048\)
\(L(\frac12)\) \(\approx\) \(4.530391048\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 - T \)
good11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−13.49772255049184, −12.99807813676280, −12.57001086530253, −11.95058555140025, −11.66977128150858, −11.03160675827804, −10.39761595804352, −10.08537831924640, −9.509087602301116, −8.993471699038173, −8.642463516130895, −8.282202868649367, −7.536496651150805, −6.894432338654341, −6.551203514956117, −6.133697265960351, −5.532028043880879, −4.659815579613389, −4.298358521896764, −3.805057000048762, −3.176713536667027, −2.476201794845248, −1.921435088445794, −1.333137584628289, −0.6304404083165849, 0.6304404083165849, 1.333137584628289, 1.921435088445794, 2.476201794845248, 3.176713536667027, 3.805057000048762, 4.298358521896764, 4.659815579613389, 5.532028043880879, 6.133697265960351, 6.551203514956117, 6.894432338654341, 7.536496651150805, 8.282202868649367, 8.642463516130895, 8.993471699038173, 9.509087602301116, 10.08537831924640, 10.39761595804352, 11.03160675827804, 11.66977128150858, 11.95058555140025, 12.57001086530253, 12.99807813676280, 13.49772255049184

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