Properties

Label 2-5225-1.1-c1-0-103
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.814·2-s − 2.50·3-s − 1.33·4-s − 2.03·6-s − 4.83·7-s − 2.71·8-s + 3.25·9-s − 11-s + 3.34·12-s − 4.12·13-s − 3.94·14-s + 0.459·16-s + 2.02·17-s + 2.64·18-s + 19-s + 12.0·21-s − 0.814·22-s + 6.38·23-s + 6.79·24-s − 3.35·26-s − 0.626·27-s + 6.46·28-s + 7.42·29-s − 4.40·31-s + 5.80·32-s + 2.50·33-s + 1.65·34-s + ⋯
L(s)  = 1  + 0.575·2-s − 1.44·3-s − 0.668·4-s − 0.831·6-s − 1.82·7-s − 0.960·8-s + 1.08·9-s − 0.301·11-s + 0.964·12-s − 1.14·13-s − 1.05·14-s + 0.114·16-s + 0.491·17-s + 0.624·18-s + 0.229·19-s + 2.64·21-s − 0.173·22-s + 1.33·23-s + 1.38·24-s − 0.658·26-s − 0.120·27-s + 1.22·28-s + 1.37·29-s − 0.791·31-s + 1.02·32-s + 0.435·33-s + 0.283·34-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: \(1\)
Selberg data: \((2,\ 5225,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 - 0.814T + 2T^{2} \)
3 \( 1 + 2.50T + 3T^{2} \)
7 \( 1 + 4.83T + 7T^{2} \)
13 \( 1 + 4.12T + 13T^{2} \)
17 \( 1 - 2.02T + 17T^{2} \)
23 \( 1 - 6.38T + 23T^{2} \)
29 \( 1 - 7.42T + 29T^{2} \)
31 \( 1 + 4.40T + 31T^{2} \)
37 \( 1 - 1.77T + 37T^{2} \)
41 \( 1 - 9.92T + 41T^{2} \)
43 \( 1 + 9.89T + 43T^{2} \)
47 \( 1 - 0.145T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 - 1.17T + 59T^{2} \)
61 \( 1 - 2.03T + 61T^{2} \)
67 \( 1 - 7.84T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 + 3.27T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 2.37T + 83T^{2} \)
89 \( 1 + 9.26T + 89T^{2} \)
97 \( 1 - 8.29T + 97T^{2} \)
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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

−7.56590066763845290540571398989, −6.76793135367799190116286176389, −6.30569109392101908574485564155, −5.56879373037343627863274909274, −5.04789490871391398694233281167, −4.38080470100731225478587947474, −3.30448962915031485311621595113, −2.78895673115108250889351394569, −0.812142640447821742607743400726, 0, 0.812142640447821742607743400726, 2.78895673115108250889351394569, 3.30448962915031485311621595113, 4.38080470100731225478587947474, 5.04789490871391398694233281167, 5.56879373037343627863274909274, 6.30569109392101908574485564155, 6.76793135367799190116286176389, 7.56590066763845290540571398989

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