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Properties

Label 2-97461-1.1-c1-0-11
Degree $2$
Conductor $97461$
Sign $1$
Analytic cond. $778.230$
Root an. cond. $27.8967$
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  − 2·2-s + 2·4-s − 2·5-s + 4·10-s + 5·11-s + 13-s − 4·16-s − 17-s + 8·19-s − 4·20-s − 10·22-s − 6·23-s − 25-s − 2·26-s + 8·29-s + 9·31-s + 8·32-s + 2·34-s + 9·37-s − 16·38-s − 7·41-s − 43-s + 10·44-s + 12·46-s + 12·47-s + 2·50-s + 2·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.894·5-s + 1.26·10-s + 1.50·11-s + 0.277·13-s − 16-s − 0.242·17-s + 1.83·19-s − 0.894·20-s − 2.13·22-s − 1.25·23-s − 1/5·25-s − 0.392·26-s + 1.48·29-s + 1.61·31-s + 1.41·32-s + 0.342·34-s + 1.47·37-s − 2.59·38-s − 1.09·41-s − 0.152·43-s + 1.50·44-s + 1.76·46-s + 1.75·47-s + 0.282·50-s + 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97461\)    =    \(3^{2} \cdot 7^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(778.230\)
Root analytic conductor: \(27.8967\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 97461,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.252373092\)
\(L(\frac12)\) \(\approx\) \(1.252373092\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
17 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−13.76181866504516, −13.70102258535680, −12.41088836903175, −12.06521845728031, −11.76463692647676, −11.33459497956454, −10.79704193227014, −10.10517766252721, −9.799419301921906, −9.266708242123968, −8.879134550649641, −8.216424484270679, −7.873527446654231, −7.539093040057806, −6.782644029171331, −6.406133686803526, −5.885411608462314, −4.831565784246148, −4.423479404503089, −3.858761294677728, −3.220165409653217, −2.501970953165690, −1.614520959241779, −1.024367918600655, −0.5666978373581314, 0.5666978373581314, 1.024367918600655, 1.614520959241779, 2.501970953165690, 3.220165409653217, 3.858761294677728, 4.423479404503089, 4.831565784246148, 5.885411608462314, 6.406133686803526, 6.782644029171331, 7.539093040057806, 7.873527446654231, 8.216424484270679, 8.879134550649641, 9.266708242123968, 9.799419301921906, 10.10517766252721, 10.79704193227014, 11.33459497956454, 11.76463692647676, 12.06521845728031, 12.41088836903175, 13.70102258535680, 13.76181866504516

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