Properties

Label 2-75e2-1.1-c1-0-70
Degree $2$
Conductor $5625$
Sign $1$
Analytic cond. $44.9158$
Root an. cond. $6.70192$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.61·2-s + 0.618·4-s + 2·7-s − 2.23·8-s + 3·11-s + 13-s + 3.23·14-s − 4.85·16-s − 4.23·17-s − 6.70·19-s + 4.85·22-s + 5.38·23-s + 1.61·26-s + 1.23·28-s + 3.61·29-s + 8.70·31-s − 3.38·32-s − 6.85·34-s + 2·37-s − 10.8·38-s + 9.38·41-s + 7.38·43-s + 1.85·44-s + 8.70·46-s − 4.76·47-s − 3·49-s + 0.618·52-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.309·4-s + 0.755·7-s − 0.790·8-s + 0.904·11-s + 0.277·13-s + 0.864·14-s − 1.21·16-s − 1.02·17-s − 1.53·19-s + 1.03·22-s + 1.12·23-s + 0.317·26-s + 0.233·28-s + 0.671·29-s + 1.56·31-s − 0.597·32-s − 1.17·34-s + 0.328·37-s − 1.76·38-s + 1.46·41-s + 1.12·43-s + 0.279·44-s + 1.28·46-s − 0.694·47-s − 0.428·49-s + 0.0857·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(44.9158\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.655648333\)
\(L(\frac12)\) \(\approx\) \(3.655648333\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 - 1.61T + 2T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + 4.23T + 17T^{2} \)
19 \( 1 + 6.70T + 19T^{2} \)
23 \( 1 - 5.38T + 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
31 \( 1 - 8.70T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 9.38T + 41T^{2} \)
43 \( 1 - 7.38T + 43T^{2} \)
47 \( 1 + 4.76T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 3.94T + 59T^{2} \)
61 \( 1 - 8.70T + 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + 15.7T + 73T^{2} \)
79 \( 1 + 9.14T + 79T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + 3.85T + 97T^{2} \)
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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

−8.296718245912122637367585276106, −7.16517218523111685404887345631, −6.42937178358094565300146826854, −6.04315260952597940359726895089, −5.00335411200033079020636110645, −4.38291838802353800698481200700, −4.07030818396533060539590693767, −2.90955978506035313830731761930, −2.16523648287972823866689000080, −0.874120518358492336014150497013, 0.874120518358492336014150497013, 2.16523648287972823866689000080, 2.90955978506035313830731761930, 4.07030818396533060539590693767, 4.38291838802353800698481200700, 5.00335411200033079020636110645, 6.04315260952597940359726895089, 6.42937178358094565300146826854, 7.16517218523111685404887345631, 8.296718245912122637367585276106

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