Properties

Label 2-5225-1.1-c1-0-107
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  − 2.15·2-s − 2.27·3-s + 2.65·4-s + 4.91·6-s + 4.96·7-s − 1.41·8-s + 2.19·9-s + 11-s − 6.05·12-s + 7.07·13-s − 10.7·14-s − 2.25·16-s − 0.882·17-s − 4.72·18-s − 19-s − 11.3·21-s − 2.15·22-s − 0.450·23-s + 3.22·24-s − 15.2·26-s + 1.84·27-s + 13.1·28-s + 4.90·29-s − 4.59·31-s + 7.70·32-s − 2.27·33-s + 1.90·34-s + ⋯
L(s)  = 1  − 1.52·2-s − 1.31·3-s + 1.32·4-s + 2.00·6-s + 1.87·7-s − 0.500·8-s + 0.730·9-s + 0.301·11-s − 1.74·12-s + 1.96·13-s − 2.86·14-s − 0.563·16-s − 0.214·17-s − 1.11·18-s − 0.229·19-s − 2.46·21-s − 0.460·22-s − 0.0940·23-s + 0.658·24-s − 2.99·26-s + 0.354·27-s + 2.49·28-s + 0.911·29-s − 0.824·31-s + 1.36·32-s − 0.396·33-s + 0.326·34-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

\(L(1)\) \(\approx\) \(0.8652455135\)
\(L(\frac12)\) \(\approx\) \(0.8652455135\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.15T + 2T^{2} \)
3 \( 1 + 2.27T + 3T^{2} \)
7 \( 1 - 4.96T + 7T^{2} \)
13 \( 1 - 7.07T + 13T^{2} \)
17 \( 1 + 0.882T + 17T^{2} \)
23 \( 1 + 0.450T + 23T^{2} \)
29 \( 1 - 4.90T + 29T^{2} \)
31 \( 1 + 4.59T + 31T^{2} \)
37 \( 1 - 9.57T + 37T^{2} \)
41 \( 1 + 3.73T + 41T^{2} \)
43 \( 1 - 6.29T + 43T^{2} \)
47 \( 1 + 5.17T + 47T^{2} \)
53 \( 1 + 3.00T + 53T^{2} \)
59 \( 1 - 7.98T + 59T^{2} \)
61 \( 1 - 2.63T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 - 6.34T + 79T^{2} \)
83 \( 1 - 17.2T + 83T^{2} \)
89 \( 1 - 5.27T + 89T^{2} \)
97 \( 1 + 19.4T + 97T^{2} \)
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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

−8.314166966787445313528628910261, −7.75779683613153308441909450694, −6.78633402170708447883213916532, −6.24563887684653981219673509190, −5.43569725594745357630027518238, −4.68251960959837681843976660949, −3.91136367870102382442300053928, −2.22709309017810230238845094178, −1.30505817085820353106357034106, −0.821326783007062777735877436847, 0.821326783007062777735877436847, 1.30505817085820353106357034106, 2.22709309017810230238845094178, 3.91136367870102382442300053928, 4.68251960959837681843976660949, 5.43569725594745357630027518238, 6.24563887684653981219673509190, 6.78633402170708447883213916532, 7.75779683613153308441909450694, 8.314166966787445313528628910261

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