Properties

Label 2-6025-1.1-c1-0-252
Degree $2$
Conductor $6025$
Sign $-1$
Analytic cond. $48.1098$
Root an. cond. $6.93612$
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  − 1.59·2-s + 0.0342·3-s + 0.533·4-s − 0.0545·6-s − 1.01·7-s + 2.33·8-s − 2.99·9-s + 5.74·11-s + 0.0182·12-s + 4.77·13-s + 1.62·14-s − 4.78·16-s − 2.20·17-s + 4.77·18-s + 7.81·19-s − 0.0348·21-s − 9.13·22-s − 3.98·23-s + 0.0800·24-s − 7.60·26-s − 0.205·27-s − 0.543·28-s + 0.409·29-s − 9.04·31-s + 2.94·32-s + 0.196·33-s + 3.51·34-s + ⋯
L(s)  = 1  − 1.12·2-s + 0.0197·3-s + 0.266·4-s − 0.0222·6-s − 0.384·7-s + 0.825·8-s − 0.999·9-s + 1.73·11-s + 0.00528·12-s + 1.32·13-s + 0.433·14-s − 1.19·16-s − 0.535·17-s + 1.12·18-s + 1.79·19-s − 0.00761·21-s − 1.94·22-s − 0.830·23-s + 0.0163·24-s − 1.49·26-s − 0.0395·27-s − 0.102·28-s + 0.0760·29-s − 1.62·31-s + 0.520·32-s + 0.0342·33-s + 0.602·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-1$
Analytic conductor: \(48.1098\)
Root analytic conductor: \(6.93612\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ -1)\)

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 \( 1 \)
241 \( 1 + T \)
good2 \( 1 + 1.59T + 2T^{2} \)
3 \( 1 - 0.0342T + 3T^{2} \)
7 \( 1 + 1.01T + 7T^{2} \)
11 \( 1 - 5.74T + 11T^{2} \)
13 \( 1 - 4.77T + 13T^{2} \)
17 \( 1 + 2.20T + 17T^{2} \)
19 \( 1 - 7.81T + 19T^{2} \)
23 \( 1 + 3.98T + 23T^{2} \)
29 \( 1 - 0.409T + 29T^{2} \)
31 \( 1 + 9.04T + 31T^{2} \)
37 \( 1 + 7.42T + 37T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 + 0.457T + 43T^{2} \)
47 \( 1 + 6.21T + 47T^{2} \)
53 \( 1 + 2.25T + 53T^{2} \)
59 \( 1 - 8.50T + 59T^{2} \)
61 \( 1 - 2.47T + 61T^{2} \)
67 \( 1 - 3.16T + 67T^{2} \)
71 \( 1 - 1.46T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 - 16.2T + 79T^{2} \)
83 \( 1 + 2.52T + 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−7.967351009617663533742187963034, −7.03816819971506628785986177850, −6.53986867071436209349816956924, −5.73136048885822582825330007207, −4.91258784221702439519822569372, −3.62826003286511059392827493033, −3.50032429284690900962620411537, −1.88595887928303975432192381460, −1.20275576882208343181983826210, 0, 1.20275576882208343181983826210, 1.88595887928303975432192381460, 3.50032429284690900962620411537, 3.62826003286511059392827493033, 4.91258784221702439519822569372, 5.73136048885822582825330007207, 6.53986867071436209349816956924, 7.03816819971506628785986177850, 7.967351009617663533742187963034

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