Properties

Label 2-99e2-1.1-c1-0-181
Degree $2$
Conductor $9801$
Sign $-1$
Analytic cond. $78.2613$
Root an. cond. $8.84654$
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  − 1.73·2-s + 0.999·4-s − 1.73·5-s − 2·7-s + 1.73·8-s + 2.99·10-s + 13-s + 3.46·14-s − 5·16-s − 5.19·17-s − 2·19-s − 1.73·20-s + 3.46·23-s − 2.00·25-s − 1.73·26-s − 1.99·28-s − 1.73·29-s + 8·31-s + 5.19·32-s + 9·34-s + 3.46·35-s − 7·37-s + 3.46·38-s − 3.00·40-s + 6.92·41-s − 2·43-s − 5.99·46-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.499·4-s − 0.774·5-s − 0.755·7-s + 0.612·8-s + 0.948·10-s + 0.277·13-s + 0.925·14-s − 1.25·16-s − 1.26·17-s − 0.458·19-s − 0.387·20-s + 0.722·23-s − 0.400·25-s − 0.339·26-s − 0.377·28-s − 0.321·29-s + 1.43·31-s + 0.918·32-s + 1.54·34-s + 0.585·35-s − 1.15·37-s + 0.561·38-s − 0.474·40-s + 1.08·41-s − 0.304·43-s − 0.884·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9801\)    =    \(3^{4} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(78.2613\)
Root analytic conductor: \(8.84654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9801,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + 1.73T + 2T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + 5.19T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−7.47602425196733180808510235490, −6.80891826212390314326490246244, −6.35559472718337603287764484755, −5.27008504006369175030698491509, −4.36039868496485172908644907565, −3.89884842017188538670384766657, −2.85719586974089316958048605023, −1.99728663955535100055940092563, −0.834913515341981323344590190895, 0, 0.834913515341981323344590190895, 1.99728663955535100055940092563, 2.85719586974089316958048605023, 3.89884842017188538670384766657, 4.36039868496485172908644907565, 5.27008504006369175030698491509, 6.35559472718337603287764484755, 6.80891826212390314326490246244, 7.47602425196733180808510235490

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