Properties

Label 2-53312-1.1-c1-0-17
Degree $2$
Conductor $53312$
Sign $-1$
Analytic cond. $425.698$
Root an. cond. $20.6324$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·5-s + 9-s + 6·11-s − 2·13-s + 8·15-s + 17-s − 4·23-s + 11·25-s + 4·27-s − 8·29-s − 12·33-s − 4·37-s + 4·39-s + 2·41-s + 8·43-s − 4·45-s + 8·47-s − 2·51-s + 6·53-s − 24·55-s − 4·59-s − 8·61-s + 8·65-s + 16·67-s + 8·69-s + 4·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 2.06·15-s + 0.242·17-s − 0.834·23-s + 11/5·25-s + 0.769·27-s − 1.48·29-s − 2.08·33-s − 0.657·37-s + 0.640·39-s + 0.312·41-s + 1.21·43-s − 0.596·45-s + 1.16·47-s − 0.280·51-s + 0.824·53-s − 3.23·55-s − 0.520·59-s − 1.02·61-s + 0.992·65-s + 1.95·67-s + 0.963·69-s + 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53312\)    =    \(2^{6} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(425.698\)
Root analytic conductor: \(20.6324\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 53312,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
17 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.68436812864008, −14.34897722117898, −13.84423667183349, −12.80692602211580, −12.38445911852601, −12.03242119334227, −11.72840113711620, −11.18360828105503, −10.94162651003039, −10.20861808651177, −9.519170918497881, −8.888140579561164, −8.537401119940145, −7.570274649315669, −7.447792686254153, −6.780290332934965, −6.223076236274977, −5.649964405445621, −4.997139522218325, −4.257232136203193, −3.968979593129054, −3.461195663046533, −2.506101775319304, −1.429973537547471, −0.7019466687430296, 0, 0.7019466687430296, 1.429973537547471, 2.506101775319304, 3.461195663046533, 3.968979593129054, 4.257232136203193, 4.997139522218325, 5.649964405445621, 6.223076236274977, 6.780290332934965, 7.447792686254153, 7.570274649315669, 8.537401119940145, 8.888140579561164, 9.519170918497881, 10.20861808651177, 10.94162651003039, 11.18360828105503, 11.72840113711620, 12.03242119334227, 12.38445911852601, 12.80692602211580, 13.84423667183349, 14.34897722117898, 14.68436812864008

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