Properties

Label 2-5225-1.1-c1-0-204
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  + 2.40·2-s + 1.17·3-s + 3.77·4-s + 2.83·6-s + 2.15·7-s + 4.26·8-s − 1.60·9-s + 11-s + 4.45·12-s + 3.54·13-s + 5.18·14-s + 2.69·16-s + 0.622·17-s − 3.86·18-s − 19-s + 2.54·21-s + 2.40·22-s + 5.67·23-s + 5.02·24-s + 8.52·26-s − 5.43·27-s + 8.13·28-s + 7.32·29-s − 7.89·31-s − 2.04·32-s + 1.17·33-s + 1.49·34-s + ⋯
L(s)  = 1  + 1.69·2-s + 0.680·3-s + 1.88·4-s + 1.15·6-s + 0.814·7-s + 1.50·8-s − 0.536·9-s + 0.301·11-s + 1.28·12-s + 0.983·13-s + 1.38·14-s + 0.674·16-s + 0.150·17-s − 0.911·18-s − 0.229·19-s + 0.554·21-s + 0.512·22-s + 1.18·23-s + 1.02·24-s + 1.67·26-s − 1.04·27-s + 1.53·28-s + 1.36·29-s − 1.41·31-s − 0.361·32-s + 0.205·33-s + 0.256·34-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

\(L(1)\) \(\approx\) \(8.092210687\)
\(L(\frac12)\) \(\approx\) \(8.092210687\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.40T + 2T^{2} \)
3 \( 1 - 1.17T + 3T^{2} \)
7 \( 1 - 2.15T + 7T^{2} \)
13 \( 1 - 3.54T + 13T^{2} \)
17 \( 1 - 0.622T + 17T^{2} \)
23 \( 1 - 5.67T + 23T^{2} \)
29 \( 1 - 7.32T + 29T^{2} \)
31 \( 1 + 7.89T + 31T^{2} \)
37 \( 1 - 9.19T + 37T^{2} \)
41 \( 1 - 2.29T + 41T^{2} \)
43 \( 1 + 0.194T + 43T^{2} \)
47 \( 1 + 0.340T + 47T^{2} \)
53 \( 1 - 6.94T + 53T^{2} \)
59 \( 1 - 8.99T + 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 - 0.226T + 67T^{2} \)
71 \( 1 - 0.343T + 71T^{2} \)
73 \( 1 + 9.18T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 + 3.15T + 83T^{2} \)
89 \( 1 - 5.44T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−8.117093035850347652345032039015, −7.35519588434228058927805756605, −6.52493341889137396035465162261, −5.86639016018693377905516405475, −5.20451960147260295145674168665, −4.44575214663105049989914019998, −3.76082173374369259299460860177, −3.03873861189817211015699771336, −2.34391640151560885927780585492, −1.29540527429513490176027886106, 1.29540527429513490176027886106, 2.34391640151560885927780585492, 3.03873861189817211015699771336, 3.76082173374369259299460860177, 4.44575214663105049989914019998, 5.20451960147260295145674168665, 5.86639016018693377905516405475, 6.52493341889137396035465162261, 7.35519588434228058927805756605, 8.117093035850347652345032039015

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