Properties

Label 2-4425-1.1-c1-0-92
Degree $2$
Conductor $4425$
Sign $1$
Analytic cond. $35.3338$
Root an. cond. $5.94422$
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  + 1.61·2-s + 3-s + 0.618·4-s + 1.61·6-s + 4.61·7-s − 2.23·8-s + 9-s − 2.23·11-s + 0.618·12-s + 1.76·13-s + 7.47·14-s − 4.85·16-s + 4.85·17-s + 1.61·18-s − 8.09·19-s + 4.61·21-s − 3.61·22-s + 2.38·23-s − 2.23·24-s + 2.85·26-s + 27-s + 2.85·28-s + 8.61·29-s + 9.56·31-s − 3.38·32-s − 2.23·33-s + 7.85·34-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.660·6-s + 1.74·7-s − 0.790·8-s + 0.333·9-s − 0.674·11-s + 0.178·12-s + 0.489·13-s + 1.99·14-s − 1.21·16-s + 1.17·17-s + 0.381·18-s − 1.85·19-s + 1.00·21-s − 0.771·22-s + 0.496·23-s − 0.456·24-s + 0.559·26-s + 0.192·27-s + 0.539·28-s + 1.60·29-s + 1.71·31-s − 0.597·32-s − 0.389·33-s + 1.34·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4425\)    =    \(3 \cdot 5^{2} \cdot 59\)
Sign: $1$
Analytic conductor: \(35.3338\)
Root analytic conductor: \(5.94422\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4425,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.959188947\)
\(L(\frac12)\) \(\approx\) \(4.959188947\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
59 \( 1 + T \)
good2 \( 1 - 1.61T + 2T^{2} \)
7 \( 1 - 4.61T + 7T^{2} \)
11 \( 1 + 2.23T + 11T^{2} \)
13 \( 1 - 1.76T + 13T^{2} \)
17 \( 1 - 4.85T + 17T^{2} \)
19 \( 1 + 8.09T + 19T^{2} \)
23 \( 1 - 2.38T + 23T^{2} \)
29 \( 1 - 8.61T + 29T^{2} \)
31 \( 1 - 9.56T + 31T^{2} \)
37 \( 1 - 6.85T + 37T^{2} \)
41 \( 1 + 3.09T + 41T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 - 4.14T + 47T^{2} \)
53 \( 1 + 1.76T + 53T^{2} \)
61 \( 1 + 9.85T + 61T^{2} \)
67 \( 1 - 2.70T + 67T^{2} \)
71 \( 1 - 9.94T + 71T^{2} \)
73 \( 1 - 5.85T + 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 + 0.618T + 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + 3T + 97T^{2} \)
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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

−8.252419121729963466604057602243, −7.895568097799451998359497492915, −6.70631292813660061906677071970, −6.01550206942327817836940518226, −5.06420405368235833901953968529, −4.65393605379834373742175500608, −4.01903409892236909916379294999, −2.96426507038382580072113218299, −2.28438336394772667286516783471, −1.10115142098212647082118293732, 1.10115142098212647082118293732, 2.28438336394772667286516783471, 2.96426507038382580072113218299, 4.01903409892236909916379294999, 4.65393605379834373742175500608, 5.06420405368235833901953968529, 6.01550206942327817836940518226, 6.70631292813660061906677071970, 7.895568097799451998359497492915, 8.252419121729963466604057602243

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