Properties

Label 2-5225-1.1-c1-0-133
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s − 2·3-s + 3.82·4-s + 4.82·6-s + 2.82·7-s − 4.41·8-s + 9-s − 11-s − 7.65·12-s − 6.82·13-s − 6.82·14-s + 2.99·16-s + 4.82·17-s − 2.41·18-s − 19-s − 5.65·21-s + 2.41·22-s − 4·23-s + 8.82·24-s + 16.4·26-s + 4·27-s + 10.8·28-s + 3.17·29-s + 1.17·31-s + 1.58·32-s + 2·33-s − 11.6·34-s + ⋯
L(s)  = 1  − 1.70·2-s − 1.15·3-s + 1.91·4-s + 1.97·6-s + 1.06·7-s − 1.56·8-s + 0.333·9-s − 0.301·11-s − 2.21·12-s − 1.89·13-s − 1.82·14-s + 0.749·16-s + 1.17·17-s − 0.569·18-s − 0.229·19-s − 1.23·21-s + 0.514·22-s − 0.834·23-s + 1.80·24-s + 3.23·26-s + 0.769·27-s + 2.04·28-s + 0.588·29-s + 0.210·31-s + 0.280·32-s + 0.348·33-s − 1.99·34-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: \(1\)
Selberg data: \((2,\ 5225,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41T + 2T^{2} \)
3 \( 1 + 2T + 3T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
13 \( 1 + 6.82T + 13T^{2} \)
17 \( 1 - 4.82T + 17T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 3.17T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 - 1.17T + 37T^{2} \)
41 \( 1 + 8.82T + 41T^{2} \)
43 \( 1 - 6.82T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 2.82T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 3.65T + 61T^{2} \)
67 \( 1 + 0.343T + 67T^{2} \)
71 \( 1 - 9.17T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−8.101211789969037062980804442935, −7.23676322552597305833779700448, −6.75287869072939062097400271647, −5.70095390374492952867567242742, −5.16885079279985434694299406497, −4.39173508761125659924094710677, −2.78810026844554095991757947336, −1.97971899917903327055959651648, −0.956520665155033845358800161461, 0, 0.956520665155033845358800161461, 1.97971899917903327055959651648, 2.78810026844554095991757947336, 4.39173508761125659924094710677, 5.16885079279985434694299406497, 5.70095390374492952867567242742, 6.75287869072939062097400271647, 7.23676322552597305833779700448, 8.101211789969037062980804442935

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