Properties

Label 2-5225-1.1-c1-0-0
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  − 0.379·2-s − 2.60·3-s − 1.85·4-s + 0.989·6-s − 4.15·7-s + 1.46·8-s + 3.79·9-s + 11-s + 4.84·12-s − 0.808·13-s + 1.57·14-s + 3.15·16-s − 5.26·17-s − 1.44·18-s + 19-s + 10.8·21-s − 0.379·22-s − 2.49·23-s − 3.81·24-s + 0.306·26-s − 2.08·27-s + 7.71·28-s − 2.90·29-s + 2.50·31-s − 4.12·32-s − 2.60·33-s + 1.99·34-s + ⋯
L(s)  = 1  − 0.268·2-s − 1.50·3-s − 0.928·4-s + 0.403·6-s − 1.57·7-s + 0.517·8-s + 1.26·9-s + 0.301·11-s + 1.39·12-s − 0.224·13-s + 0.421·14-s + 0.789·16-s − 1.27·17-s − 0.339·18-s + 0.229·19-s + 2.36·21-s − 0.0808·22-s − 0.520·23-s − 0.778·24-s + 0.0601·26-s − 0.401·27-s + 1.45·28-s − 0.539·29-s + 0.450·31-s − 0.728·32-s − 0.453·33-s + 0.342·34-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

\(L(1)\) \(\approx\) \(0.02811550649\)
\(L(\frac12)\) \(\approx\) \(0.02811550649\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.379T + 2T^{2} \)
3 \( 1 + 2.60T + 3T^{2} \)
7 \( 1 + 4.15T + 7T^{2} \)
13 \( 1 + 0.808T + 13T^{2} \)
17 \( 1 + 5.26T + 17T^{2} \)
23 \( 1 + 2.49T + 23T^{2} \)
29 \( 1 + 2.90T + 29T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 - 2.11T + 37T^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + 1.42T + 43T^{2} \)
47 \( 1 + 4.28T + 47T^{2} \)
53 \( 1 + 7.81T + 53T^{2} \)
59 \( 1 + 9.86T + 59T^{2} \)
61 \( 1 + 1.27T + 61T^{2} \)
67 \( 1 - 7.58T + 67T^{2} \)
71 \( 1 + 8.49T + 71T^{2} \)
73 \( 1 + 6.86T + 73T^{2} \)
79 \( 1 - 3.74T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 1.09T + 89T^{2} \)
97 \( 1 + 9.76T + 97T^{2} \)
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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

−8.315567740230176373997692472788, −7.25142346038322121292401234889, −6.58218692277548889001887105923, −6.13474357059829981891710740581, −5.34371819376329220667724721191, −4.62197776832814329050648908893, −3.93446115360280805351511138883, −2.98617341502962092620538621458, −1.46258589958900213051520442736, −0.10670679273180952804405928847, 0.10670679273180952804405928847, 1.46258589958900213051520442736, 2.98617341502962092620538621458, 3.93446115360280805351511138883, 4.62197776832814329050648908893, 5.34371819376329220667724721191, 6.13474357059829981891710740581, 6.58218692277548889001887105923, 7.25142346038322121292401234889, 8.315567740230176373997692472788

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