Properties

Label 2-6025-1.1-c1-0-240
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.747·2-s + 2.67·3-s − 1.44·4-s − 2.00·6-s + 3.83·7-s + 2.57·8-s + 4.16·9-s − 2.47·11-s − 3.85·12-s + 5.59·13-s − 2.86·14-s + 0.960·16-s + 5.49·17-s − 3.11·18-s + 2.62·19-s + 10.2·21-s + 1.85·22-s + 7.47·23-s + 6.88·24-s − 4.18·26-s + 3.12·27-s − 5.53·28-s − 0.734·29-s + 3.92·31-s − 5.86·32-s − 6.62·33-s − 4.10·34-s + ⋯
L(s)  = 1  − 0.528·2-s + 1.54·3-s − 0.720·4-s − 0.816·6-s + 1.45·7-s + 0.909·8-s + 1.38·9-s − 0.746·11-s − 1.11·12-s + 1.55·13-s − 0.766·14-s + 0.240·16-s + 1.33·17-s − 0.734·18-s + 0.601·19-s + 2.24·21-s + 0.394·22-s + 1.55·23-s + 1.40·24-s − 0.819·26-s + 0.601·27-s − 1.04·28-s − 0.136·29-s + 0.705·31-s − 1.03·32-s − 1.15·33-s − 0.704·34-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: \(0\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.286964164\)
\(L(\frac12)\) \(\approx\) \(3.286964164\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.747T + 2T^{2} \)
3 \( 1 - 2.67T + 3T^{2} \)
7 \( 1 - 3.83T + 7T^{2} \)
11 \( 1 + 2.47T + 11T^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
17 \( 1 - 5.49T + 17T^{2} \)
19 \( 1 - 2.62T + 19T^{2} \)
23 \( 1 - 7.47T + 23T^{2} \)
29 \( 1 + 0.734T + 29T^{2} \)
31 \( 1 - 3.92T + 31T^{2} \)
37 \( 1 - 4.79T + 37T^{2} \)
41 \( 1 + 7.98T + 41T^{2} \)
43 \( 1 + 2.88T + 43T^{2} \)
47 \( 1 + 6.57T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 - 8.06T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + 2.58T + 71T^{2} \)
73 \( 1 + 7.39T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + 9.72T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 8.19T + 97T^{2} \)
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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

−8.236520929744981350073154727845, −7.71760389432325336755867377323, −7.21665717964195020235117815120, −5.77963531785546212659462683762, −5.04442434819038621290668097417, −4.37900542673536611276713390588, −3.47330906776566429422269494796, −2.89304068391492809051035216145, −1.57160483266881709090981699378, −1.15225794640948668222228522605, 1.15225794640948668222228522605, 1.57160483266881709090981699378, 2.89304068391492809051035216145, 3.47330906776566429422269494796, 4.37900542673536611276713390588, 5.04442434819038621290668097417, 5.77963531785546212659462683762, 7.21665717964195020235117815120, 7.71760389432325336755867377323, 8.236520929744981350073154727845

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