Properties

Label 2-55825-1.1-c1-0-10
Degree $2$
Conductor $55825$
Sign $-1$
Analytic cond. $445.764$
Root an. cond. $21.1131$
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  + 2-s − 4-s + 7-s − 3·8-s − 3·9-s − 11-s − 6·13-s + 14-s − 16-s + 2·17-s − 3·18-s − 8·19-s − 22-s − 6·26-s − 28-s + 29-s − 4·31-s + 5·32-s + 2·34-s + 3·36-s + 2·37-s − 8·38-s − 2·41-s + 4·43-s + 44-s + 4·47-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 9-s − 0.301·11-s − 1.66·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s − 1.83·19-s − 0.213·22-s − 1.17·26-s − 0.188·28-s + 0.185·29-s − 0.718·31-s + 0.883·32-s + 0.342·34-s + 1/2·36-s + 0.328·37-s − 1.29·38-s − 0.312·41-s + 0.609·43-s + 0.150·44-s + 0.583·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55825\)    =    \(5^{2} \cdot 7 \cdot 11 \cdot 29\)
Sign: $-1$
Analytic conductor: \(445.764\)
Root analytic conductor: \(21.1131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 55825,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
29 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.67409345828806, −14.21850912577226, −13.69858052810866, −13.11007591339277, −12.62963029873533, −12.19469659053903, −11.82892276704355, −11.12938698152169, −10.61146169478727, −10.01033788886852, −9.508767686247038, −8.838229638494224, −8.480458368830804, −7.952507892943105, −7.262625528225798, −6.680332375371514, −5.888857667085541, −5.523897510456000, −5.011508643020376, −4.428656729289803, −3.936707874163119, −3.163064562602785, −2.448354476687644, −2.140574473020192, −0.7069924217443032, 0, 0.7069924217443032, 2.140574473020192, 2.448354476687644, 3.163064562602785, 3.936707874163119, 4.428656729289803, 5.011508643020376, 5.523897510456000, 5.888857667085541, 6.680332375371514, 7.262625528225798, 7.952507892943105, 8.480458368830804, 8.838229638494224, 9.508767686247038, 10.01033788886852, 10.61146169478727, 11.12938698152169, 11.82892276704355, 12.19469659053903, 12.62963029873533, 13.11007591339277, 13.69858052810866, 14.21850912577226, 14.67409345828806

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