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Properties

Label 2-95e2-1.1-c1-0-216
Degree $2$
Conductor $9025$
Sign $-1$
Analytic cond. $72.0649$
Root an. cond. $8.48910$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 4·7-s − 3·9-s − 11-s + 2·13-s + 8·14-s − 4·16-s − 2·17-s + 6·18-s + 2·22-s + 6·23-s − 4·26-s − 8·28-s − 9·29-s + 7·31-s + 8·32-s + 4·34-s − 6·36-s − 2·37-s − 2·41-s + 2·43-s − 2·44-s − 12·46-s + 6·47-s + 9·49-s + 4·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.51·7-s − 9-s − 0.301·11-s + 0.554·13-s + 2.13·14-s − 16-s − 0.485·17-s + 1.41·18-s + 0.426·22-s + 1.25·23-s − 0.784·26-s − 1.51·28-s − 1.67·29-s + 1.25·31-s + 1.41·32-s + 0.685·34-s − 36-s − 0.328·37-s − 0.312·41-s + 0.304·43-s − 0.301·44-s − 1.76·46-s + 0.875·47-s + 9/7·49-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(72.0649\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9025,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.51689909786797771491140391670, −6.82174019170213451453487066691, −6.28360229852540375243964692246, −5.58675723699578516917073033990, −4.61604020013260083325582315618, −3.53134444172460337429651985785, −2.91994397980323778713310415409, −2.07434307000442273481405593332, −0.823353708736406315621161374971, 0, 0.823353708736406315621161374971, 2.07434307000442273481405593332, 2.91994397980323778713310415409, 3.53134444172460337429651985785, 4.61604020013260083325582315618, 5.58675723699578516917073033990, 6.28360229852540375243964692246, 6.82174019170213451453487066691, 7.51689909786797771491140391670

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