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Properties

Label 2-105e2-1.1-c1-0-4
Degree $2$
Conductor $11025$
Sign $1$
Analytic cond. $88.0350$
Root an. cond. $9.38270$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 7·13-s + 4·16-s + 7·19-s + 7·31-s + 37-s − 5·43-s + 14·52-s − 14·61-s − 8·64-s − 11·67-s − 7·73-s − 14·76-s − 13·79-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 4-s − 1.94·13-s + 16-s + 1.60·19-s + 1.25·31-s + 0.164·37-s − 0.762·43-s + 1.94·52-s − 1.79·61-s − 64-s − 1.34·67-s − 0.819·73-s − 1.60·76-s − 1.46·79-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(88.0350\)
Root analytic conductor: \(9.38270\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 11025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.081521113\)
\(L(\frac12)\) \(\approx\) \(1.081521113\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.61258139223701, −15.96957203593803, −15.18580870696054, −14.77938123448080, −14.11791316820387, −13.75354103870537, −13.12279074842885, −12.43814889379538, −11.95908325149761, −11.52468544963293, −10.35940226029709, −10.05066267034014, −9.476358465367494, −8.994350389100878, −8.165360561047153, −7.564214068400154, −7.137679194840246, −6.100722498393278, −5.402551153050883, −4.763923625670569, −4.402141274703869, −3.260457125717232, −2.776775483143016, −1.562264018334010, −0.4974465266779612, 0.4974465266779612, 1.562264018334010, 2.776775483143016, 3.260457125717232, 4.402141274703869, 4.763923625670569, 5.402551153050883, 6.100722498393278, 7.137679194840246, 7.564214068400154, 8.165360561047153, 8.994350389100878, 9.476358465367494, 10.05066267034014, 10.35940226029709, 11.52468544963293, 11.95908325149761, 12.43814889379538, 13.12279074842885, 13.75354103870537, 14.11791316820387, 14.77938123448080, 15.18580870696054, 15.96957203593803, 16.61258139223701

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