Properties

Label 2-95e2-1.1-c1-0-286
Degree $2$
Conductor $9025$
Sign $-1$
Analytic cond. $72.0649$
Root an. cond. $8.48910$
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  + 1.19·2-s − 3.04·3-s − 0.581·4-s − 3.62·6-s + 0.609·7-s − 3.07·8-s + 6.28·9-s + 4.48·11-s + 1.77·12-s − 4.43·13-s + 0.725·14-s − 2.49·16-s − 2.90·17-s + 7.48·18-s − 1.85·21-s + 5.34·22-s + 2.84·23-s + 9.36·24-s − 5.28·26-s − 10.0·27-s − 0.354·28-s + 1.11·29-s − 6.22·31-s + 3.17·32-s − 13.6·33-s − 3.45·34-s − 3.65·36-s + ⋯
L(s)  = 1  + 0.842·2-s − 1.75·3-s − 0.290·4-s − 1.48·6-s + 0.230·7-s − 1.08·8-s + 2.09·9-s + 1.35·11-s + 0.511·12-s − 1.23·13-s + 0.193·14-s − 0.624·16-s − 0.704·17-s + 1.76·18-s − 0.405·21-s + 1.13·22-s + 0.593·23-s + 1.91·24-s − 1.03·26-s − 1.92·27-s − 0.0669·28-s + 0.207·29-s − 1.11·31-s + 0.561·32-s − 2.37·33-s − 0.592·34-s − 0.609·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(72.0649\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9025,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 - 1.19T + 2T^{2} \)
3 \( 1 + 3.04T + 3T^{2} \)
7 \( 1 - 0.609T + 7T^{2} \)
11 \( 1 - 4.48T + 11T^{2} \)
13 \( 1 + 4.43T + 13T^{2} \)
17 \( 1 + 2.90T + 17T^{2} \)
23 \( 1 - 2.84T + 23T^{2} \)
29 \( 1 - 1.11T + 29T^{2} \)
31 \( 1 + 6.22T + 31T^{2} \)
37 \( 1 - 3.77T + 37T^{2} \)
41 \( 1 + 8.30T + 41T^{2} \)
43 \( 1 - 9.98T + 43T^{2} \)
47 \( 1 - 5.88T + 47T^{2} \)
53 \( 1 + 8.44T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 4.98T + 61T^{2} \)
67 \( 1 + 8.47T + 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 3.72T + 73T^{2} \)
79 \( 1 - 9.03T + 79T^{2} \)
83 \( 1 - 2.12T + 83T^{2} \)
89 \( 1 - 7.93T + 89T^{2} \)
97 \( 1 - 9.67T + 97T^{2} \)
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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

−6.95459071599063956501452436559, −6.55410683023883239063004793590, −5.88135117019172350515033802083, −5.24395247452301827089219220754, −4.67939392550295180280304779779, −4.25473965588846412516485954432, −3.40507196967365482636712075711, −2.13478491654600026383660495523, −0.984566686385143110421066390348, 0, 0.984566686385143110421066390348, 2.13478491654600026383660495523, 3.40507196967365482636712075711, 4.25473965588846412516485954432, 4.67939392550295180280304779779, 5.24395247452301827089219220754, 5.88135117019172350515033802083, 6.55410683023883239063004793590, 6.95459071599063956501452436559

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