Properties

Label 2-4025-1.1-c1-0-7
Degree $2$
Conductor $4025$
Sign $1$
Analytic cond. $32.1397$
Root an. cond. $5.66919$
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  − 2.33·2-s + 0.00757·3-s + 3.46·4-s − 0.0177·6-s + 7-s − 3.41·8-s − 2.99·9-s − 4.99·11-s + 0.0262·12-s − 3.87·13-s − 2.33·14-s + 1.05·16-s − 0.475·17-s + 7.01·18-s − 1.54·19-s + 0.00757·21-s + 11.6·22-s + 23-s − 0.0258·24-s + 9.06·26-s − 0.0454·27-s + 3.46·28-s − 9.38·29-s − 11.0·31-s + 4.35·32-s − 0.0378·33-s + 1.11·34-s + ⋯
L(s)  = 1  − 1.65·2-s + 0.00437·3-s + 1.73·4-s − 0.00722·6-s + 0.377·7-s − 1.20·8-s − 0.999·9-s − 1.50·11-s + 0.00756·12-s − 1.07·13-s − 0.624·14-s + 0.264·16-s − 0.115·17-s + 1.65·18-s − 0.353·19-s + 0.00165·21-s + 2.48·22-s + 0.208·23-s − 0.00528·24-s + 1.77·26-s − 0.00874·27-s + 0.654·28-s − 1.74·29-s − 1.99·31-s + 0.770·32-s − 0.00658·33-s + 0.190·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(32.1397\)
Root analytic conductor: \(5.66919\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2272222847\)
\(L(\frac12)\) \(\approx\) \(0.2272222847\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 - T \)
23 \( 1 - T \)
good2 \( 1 + 2.33T + 2T^{2} \)
3 \( 1 - 0.00757T + 3T^{2} \)
11 \( 1 + 4.99T + 11T^{2} \)
13 \( 1 + 3.87T + 13T^{2} \)
17 \( 1 + 0.475T + 17T^{2} \)
19 \( 1 + 1.54T + 19T^{2} \)
29 \( 1 + 9.38T + 29T^{2} \)
31 \( 1 + 11.0T + 31T^{2} \)
37 \( 1 + 4.42T + 37T^{2} \)
41 \( 1 - 3.21T + 41T^{2} \)
43 \( 1 - 3.40T + 43T^{2} \)
47 \( 1 + 8.07T + 47T^{2} \)
53 \( 1 + 1.46T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 - 8.24T + 61T^{2} \)
67 \( 1 - 4.54T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 + 6.08T + 89T^{2} \)
97 \( 1 - 3.91T + 97T^{2} \)
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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

−8.395733370052473339882017038962, −7.79059403340463253139359731656, −7.45660867665074820245904176674, −6.53663525806260511461550065183, −5.45185323129680144503914829990, −5.02127428194839033577167737633, −3.53493555756122376112821514929, −2.42118811708395688887119073059, −1.96702533262071546680335582997, −0.32492387752611122890980391350, 0.32492387752611122890980391350, 1.96702533262071546680335582997, 2.42118811708395688887119073059, 3.53493555756122376112821514929, 5.02127428194839033577167737633, 5.45185323129680144503914829990, 6.53663525806260511461550065183, 7.45660867665074820245904176674, 7.79059403340463253139359731656, 8.395733370052473339882017038962

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