Properties

Label 2-5225-1.1-c1-0-112
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  − 0.414·2-s + 2·3-s − 1.82·4-s − 0.828·6-s − 0.828·7-s + 1.58·8-s + 9-s − 11-s − 3.65·12-s + 6.82·13-s + 0.343·14-s + 3·16-s + 6.82·17-s − 0.414·18-s − 19-s − 1.65·21-s + 0.414·22-s + 7.65·23-s + 3.17·24-s − 2.82·26-s − 4·27-s + 1.51·28-s − 4.82·29-s − 6.82·31-s − 4.41·32-s − 2·33-s − 2.82·34-s + ⋯
L(s)  = 1  − 0.292·2-s + 1.15·3-s − 0.914·4-s − 0.338·6-s − 0.313·7-s + 0.560·8-s + 0.333·9-s − 0.301·11-s − 1.05·12-s + 1.89·13-s + 0.0917·14-s + 0.750·16-s + 1.65·17-s − 0.0976·18-s − 0.229·19-s − 0.361·21-s + 0.0883·22-s + 1.59·23-s + 0.647·24-s − 0.554·26-s − 0.769·27-s + 0.286·28-s − 0.896·29-s − 1.22·31-s − 0.780·32-s − 0.348·33-s − 0.485·34-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

\(L(1)\) \(\approx\) \(2.184099353\)
\(L(\frac12)\) \(\approx\) \(2.184099353\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 2T^{2} \)
3 \( 1 - 2T + 3T^{2} \)
7 \( 1 + 0.828T + 7T^{2} \)
13 \( 1 - 6.82T + 13T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
23 \( 1 - 7.65T + 23T^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 + 6.82T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 - 6.48T + 41T^{2} \)
43 \( 1 + 0.828T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 6T + 67T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 - 1.17T + 73T^{2} \)
79 \( 1 - 5.65T + 79T^{2} \)
83 \( 1 + 6.48T + 83T^{2} \)
89 \( 1 + 4.34T + 89T^{2} \)
97 \( 1 - 1.17T + 97T^{2} \)
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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

−8.407968544762128207870746696615, −7.67048819542802411193527536035, −7.13211028149963843674632933052, −5.77138174958868088634509265896, −5.49285829217255491681055624698, −4.24584050568732101655107292456, −3.45180688397106857958541660967, −3.21434690418315461855601729320, −1.79257542867301516220392757899, −0.838085540289793619724571639420, 0.838085540289793619724571639420, 1.79257542867301516220392757899, 3.21434690418315461855601729320, 3.45180688397106857958541660967, 4.24584050568732101655107292456, 5.49285829217255491681055624698, 5.77138174958868088634509265896, 7.13211028149963843674632933052, 7.67048819542802411193527536035, 8.407968544762128207870746696615

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