Properties

Label 2-5225-1.1-c1-0-108
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  + 2.77·2-s − 2.74·3-s + 5.69·4-s − 7.61·6-s − 3.20·7-s + 10.2·8-s + 4.53·9-s + 11-s − 15.6·12-s − 3.87·13-s − 8.90·14-s + 17.0·16-s − 5.86·17-s + 12.5·18-s + 19-s + 8.80·21-s + 2.77·22-s + 6.11·23-s − 28.1·24-s − 10.7·26-s − 4.20·27-s − 18.2·28-s + 6.40·29-s + 1.66·31-s + 26.8·32-s − 2.74·33-s − 16.2·34-s + ⋯
L(s)  = 1  + 1.96·2-s − 1.58·3-s + 2.84·4-s − 3.10·6-s − 1.21·7-s + 3.62·8-s + 1.51·9-s + 0.301·11-s − 4.51·12-s − 1.07·13-s − 2.37·14-s + 4.26·16-s − 1.42·17-s + 2.96·18-s + 0.229·19-s + 1.92·21-s + 0.591·22-s + 1.27·23-s − 5.74·24-s − 2.10·26-s − 0.808·27-s − 3.45·28-s + 1.18·29-s + 0.298·31-s + 4.74·32-s − 0.477·33-s − 2.79·34-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

\(L(1)\) \(\approx\) \(3.699314609\)
\(L(\frac12)\) \(\approx\) \(3.699314609\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.77T + 2T^{2} \)
3 \( 1 + 2.74T + 3T^{2} \)
7 \( 1 + 3.20T + 7T^{2} \)
13 \( 1 + 3.87T + 13T^{2} \)
17 \( 1 + 5.86T + 17T^{2} \)
23 \( 1 - 6.11T + 23T^{2} \)
29 \( 1 - 6.40T + 29T^{2} \)
31 \( 1 - 1.66T + 31T^{2} \)
37 \( 1 + 0.251T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 2.17T + 43T^{2} \)
47 \( 1 + 2.68T + 47T^{2} \)
53 \( 1 - 8.89T + 53T^{2} \)
59 \( 1 + 4.09T + 59T^{2} \)
61 \( 1 - 4.52T + 61T^{2} \)
67 \( 1 + 2.61T + 67T^{2} \)
71 \( 1 + 5.51T + 71T^{2} \)
73 \( 1 - 2.73T + 73T^{2} \)
79 \( 1 - 8.17T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + 1.84T + 89T^{2} \)
97 \( 1 - 2.65T + 97T^{2} \)
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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

−7.49739991138455567339151064249, −6.96145000446871796909787326350, −6.39547175403154916021336455113, −6.09119542381987864051893593948, −5.13166658464516673952877642016, −4.73840469011155348831363761693, −4.04553250076866397408304322222, −3.02776282893712650897935171297, −2.30812980012559586514951786046, −0.823122583731826680011024987330, 0.823122583731826680011024987330, 2.30812980012559586514951786046, 3.02776282893712650897935171297, 4.04553250076866397408304322222, 4.73840469011155348831363761693, 5.13166658464516673952877642016, 6.09119542381987864051893593948, 6.39547175403154916021336455113, 6.96145000446871796909787326350, 7.49739991138455567339151064249

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