Properties

Label 2-5225-1.1-c1-0-114
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  − 0.202·2-s − 2.47·3-s − 1.95·4-s + 0.501·6-s + 5.06·7-s + 0.801·8-s + 3.14·9-s + 11-s + 4.85·12-s − 1.72·13-s − 1.02·14-s + 3.75·16-s + 2.65·17-s − 0.636·18-s + 19-s − 12.5·21-s − 0.202·22-s + 6.75·23-s − 1.98·24-s + 0.349·26-s − 0.357·27-s − 9.91·28-s + 10.1·29-s + 6.13·31-s − 2.36·32-s − 2.47·33-s − 0.536·34-s + ⋯
L(s)  = 1  − 0.143·2-s − 1.43·3-s − 0.979·4-s + 0.204·6-s + 1.91·7-s + 0.283·8-s + 1.04·9-s + 0.301·11-s + 1.40·12-s − 0.479·13-s − 0.273·14-s + 0.938·16-s + 0.642·17-s − 0.149·18-s + 0.229·19-s − 2.73·21-s − 0.0431·22-s + 1.40·23-s − 0.405·24-s + 0.0686·26-s − 0.0687·27-s − 1.87·28-s + 1.88·29-s + 1.10·31-s − 0.417·32-s − 0.431·33-s − 0.0920·34-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

\(L(1)\) \(\approx\) \(1.297329078\)
\(L(\frac12)\) \(\approx\) \(1.297329078\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.202T + 2T^{2} \)
3 \( 1 + 2.47T + 3T^{2} \)
7 \( 1 - 5.06T + 7T^{2} \)
13 \( 1 + 1.72T + 13T^{2} \)
17 \( 1 - 2.65T + 17T^{2} \)
23 \( 1 - 6.75T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 - 6.13T + 31T^{2} \)
37 \( 1 - 7.00T + 37T^{2} \)
41 \( 1 + 3.72T + 41T^{2} \)
43 \( 1 - 0.935T + 43T^{2} \)
47 \( 1 - 3.29T + 47T^{2} \)
53 \( 1 + 0.668T + 53T^{2} \)
59 \( 1 + 3.35T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 0.833T + 71T^{2} \)
73 \( 1 + 1.38T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 + 1.51T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−8.257740559831736832546903733874, −7.53758096011261567762010367230, −6.74944072407619006897493496530, −5.77238131325247171921334257106, −5.16313828980269990006489268288, −4.71343702867517830050072666171, −4.25155868487332971539759009830, −2.80574676737197162920418063112, −1.27658935484340235830902160315, −0.846773616267994399065315735007, 0.846773616267994399065315735007, 1.27658935484340235830902160315, 2.80574676737197162920418063112, 4.25155868487332971539759009830, 4.71343702867517830050072666171, 5.16313828980269990006489268288, 5.77238131325247171921334257106, 6.74944072407619006897493496530, 7.53758096011261567762010367230, 8.257740559831736832546903733874

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