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Properties

Label 2-97461-1.1-c1-0-7
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-s − 4-s − 2·5-s − 3·8-s − 2·10-s + 4·11-s − 13-s − 16-s + 17-s + 4·19-s + 2·20-s + 4·22-s − 25-s − 26-s − 6·29-s + 5·32-s + 34-s − 2·37-s + 4·38-s + 6·40-s + 2·41-s + 12·43-s − 4·44-s − 50-s + 52-s + 10·53-s − 8·55-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s + 1.20·11-s − 0.277·13-s − 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.196·26-s − 1.11·29-s + 0.883·32-s + 0.171·34-s − 0.328·37-s + 0.648·38-s + 0.948·40-s + 0.312·41-s + 1.82·43-s − 0.603·44-s − 0.141·50-s + 0.138·52-s + 1.37·53-s − 1.07·55-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

\(L(1)\) \(\approx\) \(2.349016518\)
\(L(\frac12)\) \(\approx\) \(2.349016518\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.79963017058928, −13.32099386286570, −12.79335219886604, −12.32037985203616, −11.81909899285652, −11.57715540408394, −11.12516197550300, −10.22468921470323, −9.816047589498882, −9.217108427206204, −8.866063210878605, −8.360442890683431, −7.582590014120083, −7.315548910488233, −6.694385521415467, −5.894793555690674, −5.610430338097723, −4.980868333551877, −4.257386555344263, −3.915797416190229, −3.578804155982704, −2.863201387611626, −2.101015364350891, −1.087063131728366, −0.5051958469432622, 0.5051958469432622, 1.087063131728366, 2.101015364350891, 2.863201387611626, 3.578804155982704, 3.915797416190229, 4.257386555344263, 4.980868333551877, 5.610430338097723, 5.894793555690674, 6.694385521415467, 7.315548910488233, 7.582590014120083, 8.360442890683431, 8.866063210878605, 9.217108427206204, 9.816047589498882, 10.22468921470323, 11.12516197550300, 11.57715540408394, 11.81909899285652, 12.32037985203616, 12.79335219886604, 13.32099386286570, 13.79963017058928

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