Properties

Label 2-5225-1.1-c1-0-138
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  + 0.779·2-s + 2.98·3-s − 1.39·4-s + 2.32·6-s − 1.06·7-s − 2.64·8-s + 5.88·9-s + 11-s − 4.14·12-s + 0.0563·13-s − 0.832·14-s + 0.720·16-s + 4.53·17-s + 4.58·18-s − 19-s − 3.18·21-s + 0.779·22-s + 1.07·23-s − 7.88·24-s + 0.0439·26-s + 8.59·27-s + 1.48·28-s + 0.299·29-s + 9.18·31-s + 5.85·32-s + 2.98·33-s + 3.53·34-s + ⋯
L(s)  = 1  + 0.551·2-s + 1.72·3-s − 0.695·4-s + 0.948·6-s − 0.403·7-s − 0.935·8-s + 1.96·9-s + 0.301·11-s − 1.19·12-s + 0.0156·13-s − 0.222·14-s + 0.180·16-s + 1.10·17-s + 1.08·18-s − 0.229·19-s − 0.694·21-s + 0.166·22-s + 0.225·23-s − 1.60·24-s + 0.00861·26-s + 1.65·27-s + 0.280·28-s + 0.0556·29-s + 1.64·31-s + 1.03·32-s + 0.518·33-s + 0.606·34-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

\(L(1)\) \(\approx\) \(4.142456440\)
\(L(\frac12)\) \(\approx\) \(4.142456440\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.779T + 2T^{2} \)
3 \( 1 - 2.98T + 3T^{2} \)
7 \( 1 + 1.06T + 7T^{2} \)
13 \( 1 - 0.0563T + 13T^{2} \)
17 \( 1 - 4.53T + 17T^{2} \)
23 \( 1 - 1.07T + 23T^{2} \)
29 \( 1 - 0.299T + 29T^{2} \)
31 \( 1 - 9.18T + 31T^{2} \)
37 \( 1 + 4.50T + 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + 2.89T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 + 1.14T + 59T^{2} \)
61 \( 1 + 8.09T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 13.9T + 83T^{2} \)
89 \( 1 + 0.183T + 89T^{2} \)
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

−8.241241491397709237845446225508, −7.77117394733287622218213255839, −6.80461401413695487973723713268, −6.06152749438417844680255585344, −5.02811174747045124223422689896, −4.33386948081024750066131602125, −3.49719443742333774873125336578, −3.17194334125720527362686265134, −2.23008775285929655213006175511, −0.972954146292420588465780984368, 0.972954146292420588465780984368, 2.23008775285929655213006175511, 3.17194334125720527362686265134, 3.49719443742333774873125336578, 4.33386948081024750066131602125, 5.02811174747045124223422689896, 6.06152749438417844680255585344, 6.80461401413695487973723713268, 7.77117394733287622218213255839, 8.241241491397709237845446225508

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