Properties

Label 2-6025-1.1-c1-0-265
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  + 2.74·2-s + 0.0877·3-s + 5.52·4-s + 0.240·6-s + 1.38·7-s + 9.67·8-s − 2.99·9-s − 1.68·11-s + 0.484·12-s + 1.34·13-s + 3.80·14-s + 15.4·16-s + 3.06·17-s − 8.20·18-s + 8.07·19-s + 0.121·21-s − 4.62·22-s + 2.18·23-s + 0.848·24-s + 3.69·26-s − 0.525·27-s + 7.67·28-s − 8.72·29-s + 7.09·31-s + 23.1·32-s − 0.147·33-s + 8.41·34-s + ⋯
L(s)  = 1  + 1.94·2-s + 0.0506·3-s + 2.76·4-s + 0.0982·6-s + 0.524·7-s + 3.42·8-s − 0.997·9-s − 0.508·11-s + 0.139·12-s + 0.373·13-s + 1.01·14-s + 3.87·16-s + 0.743·17-s − 1.93·18-s + 1.85·19-s + 0.0265·21-s − 0.986·22-s + 0.455·23-s + 0.173·24-s + 0.723·26-s − 0.101·27-s + 1.45·28-s − 1.61·29-s + 1.27·31-s + 4.09·32-s − 0.0257·33-s + 1.44·34-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

\(L(1)\) \(\approx\) \(8.362565572\)
\(L(\frac12)\) \(\approx\) \(8.362565572\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.74T + 2T^{2} \)
3 \( 1 - 0.0877T + 3T^{2} \)
7 \( 1 - 1.38T + 7T^{2} \)
11 \( 1 + 1.68T + 11T^{2} \)
13 \( 1 - 1.34T + 13T^{2} \)
17 \( 1 - 3.06T + 17T^{2} \)
19 \( 1 - 8.07T + 19T^{2} \)
23 \( 1 - 2.18T + 23T^{2} \)
29 \( 1 + 8.72T + 29T^{2} \)
31 \( 1 - 7.09T + 31T^{2} \)
37 \( 1 - 0.976T + 37T^{2} \)
41 \( 1 - 3.27T + 41T^{2} \)
43 \( 1 + 8.18T + 43T^{2} \)
47 \( 1 + 8.16T + 47T^{2} \)
53 \( 1 - 2.61T + 53T^{2} \)
59 \( 1 - 2.45T + 59T^{2} \)
61 \( 1 + 3.69T + 61T^{2} \)
67 \( 1 + 5.99T + 67T^{2} \)
71 \( 1 - 9.10T + 71T^{2} \)
73 \( 1 + 4.03T + 73T^{2} \)
79 \( 1 + 0.0889T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 - 0.520T + 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−7.73210815584656109537918992893, −7.29836865089081023396866278285, −6.28855346144780669301535298623, −5.71265919718386329043466807968, −5.15885819214785430906461699289, −4.66226782369405731330378580821, −3.37816592005364724335776840677, −3.27360207020829919769945207249, −2.25986582712691220687309921415, −1.24369350431582714728225722955, 1.24369350431582714728225722955, 2.25986582712691220687309921415, 3.27360207020829919769945207249, 3.37816592005364724335776840677, 4.66226782369405731330378580821, 5.15885819214785430906461699289, 5.71265919718386329043466807968, 6.28855346144780669301535298623, 7.29836865089081023396866278285, 7.73210815584656109537918992893

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