Properties

Label 2-55e2-1.1-c1-0-11
Degree $2$
Conductor $3025$
Sign $1$
Analytic cond. $24.1547$
Root an. cond. $4.91474$
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  − 1.23·2-s + 0.363·3-s − 0.477·4-s − 0.449·6-s − 2.58·7-s + 3.05·8-s − 2.86·9-s − 0.173·12-s − 2.75·13-s + 3.19·14-s − 2.81·16-s + 3.85·17-s + 3.53·18-s − 0.277·19-s − 0.941·21-s − 8.40·23-s + 1.11·24-s + 3.40·26-s − 2.13·27-s + 1.23·28-s + 3.32·29-s + 0.564·31-s − 2.63·32-s − 4.75·34-s + 1.36·36-s − 0.522·37-s + 0.342·38-s + ⋯
L(s)  = 1  − 0.872·2-s + 0.210·3-s − 0.238·4-s − 0.183·6-s − 0.977·7-s + 1.08·8-s − 0.955·9-s − 0.0501·12-s − 0.765·13-s + 0.852·14-s − 0.704·16-s + 0.934·17-s + 0.834·18-s − 0.0636·19-s − 0.205·21-s − 1.75·23-s + 0.227·24-s + 0.667·26-s − 0.411·27-s + 0.233·28-s + 0.617·29-s + 0.101·31-s − 0.466·32-s − 0.815·34-s + 0.228·36-s − 0.0859·37-s + 0.0555·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(24.1547\)
Root analytic conductor: \(4.91474\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4930945842\)
\(L(\frac12)\) \(\approx\) \(0.4930945842\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + 1.23T + 2T^{2} \)
3 \( 1 - 0.363T + 3T^{2} \)
7 \( 1 + 2.58T + 7T^{2} \)
13 \( 1 + 2.75T + 13T^{2} \)
17 \( 1 - 3.85T + 17T^{2} \)
19 \( 1 + 0.277T + 19T^{2} \)
23 \( 1 + 8.40T + 23T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 - 0.564T + 31T^{2} \)
37 \( 1 + 0.522T + 37T^{2} \)
41 \( 1 + 5.11T + 41T^{2} \)
43 \( 1 - 2.54T + 43T^{2} \)
47 \( 1 + 4.92T + 47T^{2} \)
53 \( 1 + 8.72T + 53T^{2} \)
59 \( 1 - 7.50T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 3.20T + 67T^{2} \)
71 \( 1 + 8.40T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 - 9.70T + 79T^{2} \)
83 \( 1 - 3.29T + 83T^{2} \)
89 \( 1 - 2.48T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−8.665898859786933237272351881601, −8.150044606420095317052603048432, −7.49504190725843631216566402060, −6.56013727047276331620415298725, −5.75666102769790132850260739050, −4.90184307159470702278358140547, −3.84806639144419988667357572253, −3.02534492552064555196025035788, −1.95828588352826110320918324804, −0.46590008332972863942747306324, 0.46590008332972863942747306324, 1.95828588352826110320918324804, 3.02534492552064555196025035788, 3.84806639144419988667357572253, 4.90184307159470702278358140547, 5.75666102769790132850260739050, 6.56013727047276331620415298725, 7.49504190725843631216566402060, 8.150044606420095317052603048432, 8.665898859786933237272351881601

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