Properties

Label 2-5225-1.1-c1-0-14
Degree $2$
Conductor $5225$
Sign $1$
Analytic cond. $41.7218$
Root an. cond. $6.45924$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.10·2-s − 1.12·3-s − 0.787·4-s − 1.24·6-s − 1.81·7-s − 3.06·8-s − 1.73·9-s + 11-s + 0.886·12-s − 3.00·13-s − 2.00·14-s − 1.80·16-s − 0.396·17-s − 1.90·18-s − 19-s + 2.04·21-s + 1.10·22-s − 2.47·23-s + 3.45·24-s − 3.31·26-s + 5.32·27-s + 1.43·28-s − 4.66·29-s − 10.5·31-s + 4.15·32-s − 1.12·33-s − 0.436·34-s + ⋯
L(s)  = 1  + 0.778·2-s − 0.650·3-s − 0.393·4-s − 0.506·6-s − 0.686·7-s − 1.08·8-s − 0.577·9-s + 0.301·11-s + 0.255·12-s − 0.833·13-s − 0.534·14-s − 0.451·16-s − 0.0960·17-s − 0.449·18-s − 0.229·19-s + 0.446·21-s + 0.234·22-s − 0.515·23-s + 0.705·24-s − 0.649·26-s + 1.02·27-s + 0.270·28-s − 0.866·29-s − 1.90·31-s + 0.733·32-s − 0.196·33-s − 0.0748·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5225\)    =    \(5^{2} \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(41.7218\)
Root analytic conductor: \(6.45924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5914622649\)
\(L(\frac12)\) \(\approx\) \(0.5914622649\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 - 1.10T + 2T^{2} \)
3 \( 1 + 1.12T + 3T^{2} \)
7 \( 1 + 1.81T + 7T^{2} \)
13 \( 1 + 3.00T + 13T^{2} \)
17 \( 1 + 0.396T + 17T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + 4.66T + 29T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 - 3.18T + 37T^{2} \)
41 \( 1 + 9.74T + 41T^{2} \)
43 \( 1 - 3.52T + 43T^{2} \)
47 \( 1 + 3.07T + 47T^{2} \)
53 \( 1 - 1.32T + 53T^{2} \)
59 \( 1 + 6.08T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 - 9.86T + 67T^{2} \)
71 \( 1 - 3.14T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 2.48T + 79T^{2} \)
83 \( 1 + 8.40T + 83T^{2} \)
89 \( 1 + 4.18T + 89T^{2} \)
97 \( 1 - 5.66T + 97T^{2} \)
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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

−8.270147555644623033497339928341, −7.24507085639014016450589510222, −6.54741993249626951913549204647, −5.82968430959716861734769543515, −5.36335852840322909277633580492, −4.62740721902219108648148082920, −3.75385733425351102355655918063, −3.13468930779850516173926010666, −2.07134974930221152501732651857, −0.36054296666983698895810426076, 0.36054296666983698895810426076, 2.07134974930221152501732651857, 3.13468930779850516173926010666, 3.75385733425351102355655918063, 4.62740721902219108648148082920, 5.36335852840322909277633580492, 5.82968430959716861734769543515, 6.54741993249626951913549204647, 7.24507085639014016450589510222, 8.270147555644623033497339928341

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