Properties

Label 2-5225-1.1-c1-0-11
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  − 2.03·2-s − 1.87·3-s + 2.15·4-s + 3.81·6-s − 1.92·7-s − 0.314·8-s + 0.507·9-s − 11-s − 4.03·12-s − 2.85·13-s + 3.92·14-s − 3.66·16-s + 2.33·17-s − 1.03·18-s + 19-s + 3.60·21-s + 2.03·22-s + 2.74·23-s + 0.588·24-s + 5.81·26-s + 4.66·27-s − 4.14·28-s − 0.972·29-s − 0.00551·31-s + 8.10·32-s + 1.87·33-s − 4.74·34-s + ⋯
L(s)  = 1  − 1.44·2-s − 1.08·3-s + 1.07·4-s + 1.55·6-s − 0.726·7-s − 0.111·8-s + 0.169·9-s − 0.301·11-s − 1.16·12-s − 0.790·13-s + 1.04·14-s − 0.916·16-s + 0.565·17-s − 0.243·18-s + 0.229·19-s + 0.786·21-s + 0.434·22-s + 0.572·23-s + 0.120·24-s + 1.13·26-s + 0.898·27-s − 0.783·28-s − 0.180·29-s − 0.000989·31-s + 1.43·32-s + 0.326·33-s − 0.814·34-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

\(L(1)\) \(\approx\) \(0.2016961062\)
\(L(\frac12)\) \(\approx\) \(0.2016961062\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.03T + 2T^{2} \)
3 \( 1 + 1.87T + 3T^{2} \)
7 \( 1 + 1.92T + 7T^{2} \)
13 \( 1 + 2.85T + 13T^{2} \)
17 \( 1 - 2.33T + 17T^{2} \)
23 \( 1 - 2.74T + 23T^{2} \)
29 \( 1 + 0.972T + 29T^{2} \)
31 \( 1 + 0.00551T + 31T^{2} \)
37 \( 1 + 9.67T + 37T^{2} \)
41 \( 1 - 6.65T + 41T^{2} \)
43 \( 1 + 7.99T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 - 3.74T + 61T^{2} \)
67 \( 1 - 3.97T + 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 1.87T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 - 7.57T + 97T^{2} \)
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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

−8.112814795902207918782458274759, −7.66301898290816200274902384041, −6.67908157232295934778724062230, −6.46836030493467430321935955152, −5.25213042877619843819216550935, −4.92679134378644718885176299345, −3.54305444108268131969991759067, −2.59017524957832229234425730544, −1.42254798676236773362530402399, −0.33551489458549249587905041396, 0.33551489458549249587905041396, 1.42254798676236773362530402399, 2.59017524957832229234425730544, 3.54305444108268131969991759067, 4.92679134378644718885176299345, 5.25213042877619843819216550935, 6.46836030493467430321935955152, 6.67908157232295934778724062230, 7.66301898290816200274902384041, 8.112814795902207918782458274759

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