Properties

Label 2-65e2-1.1-c1-0-187
Degree $2$
Conductor $4225$
Sign $-1$
Analytic cond. $33.7367$
Root an. cond. $5.80833$
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  − 0.801·2-s + 2.24·3-s − 1.35·4-s − 1.80·6-s + 2.35·7-s + 2.69·8-s + 2.04·9-s − 4.24·11-s − 3.04·12-s − 1.89·14-s + 0.554·16-s − 2.15·17-s − 1.64·18-s − 0.0881·19-s + 5.29·21-s + 3.40·22-s − 1.49·23-s + 6.04·24-s − 2.13·27-s − 3.19·28-s + 4.63·29-s − 6.63·31-s − 5.82·32-s − 9.54·33-s + 1.73·34-s − 2.78·36-s − 5.69·37-s + ⋯
L(s)  = 1  − 0.567·2-s + 1.29·3-s − 0.678·4-s − 0.735·6-s + 0.890·7-s + 0.951·8-s + 0.682·9-s − 1.28·11-s − 0.880·12-s − 0.505·14-s + 0.138·16-s − 0.523·17-s − 0.387·18-s − 0.0202·19-s + 1.15·21-s + 0.726·22-s − 0.311·23-s + 1.23·24-s − 0.411·27-s − 0.604·28-s + 0.859·29-s − 1.19·31-s − 1.03·32-s − 1.66·33-s + 0.296·34-s − 0.463·36-s − 0.935·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(33.7367\)
Root analytic conductor: \(5.80833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4225,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 \)
good2 \( 1 + 0.801T + 2T^{2} \)
3 \( 1 - 2.24T + 3T^{2} \)
7 \( 1 - 2.35T + 7T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
17 \( 1 + 2.15T + 17T^{2} \)
19 \( 1 + 0.0881T + 19T^{2} \)
23 \( 1 + 1.49T + 23T^{2} \)
29 \( 1 - 4.63T + 29T^{2} \)
31 \( 1 + 6.63T + 31T^{2} \)
37 \( 1 + 5.69T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 - 0.295T + 43T^{2} \)
47 \( 1 - 7.35T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 6.78T + 59T^{2} \)
61 \( 1 - 3.47T + 61T^{2} \)
67 \( 1 + 7.67T + 67T^{2} \)
71 \( 1 + 8.66T + 71T^{2} \)
73 \( 1 + 6.73T + 73T^{2} \)
79 \( 1 - 9.97T + 79T^{2} \)
83 \( 1 + 1.60T + 83T^{2} \)
89 \( 1 + 2.88T + 89T^{2} \)
97 \( 1 - 8.05T + 97T^{2} \)
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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

−8.145995951969155484302040770432, −7.69983288968865833284592401980, −7.01834046278706347095963518245, −5.59807495080066360045455585915, −4.95652903503723376873710466999, −4.18312010595867622585968864635, −3.30508706749895400716166211155, −2.33520106277708055402143235483, −1.56200769997718967540928507780, 0, 1.56200769997718967540928507780, 2.33520106277708055402143235483, 3.30508706749895400716166211155, 4.18312010595867622585968864635, 4.95652903503723376873710466999, 5.59807495080066360045455585915, 7.01834046278706347095963518245, 7.69983288968865833284592401980, 8.145995951969155484302040770432

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