Properties

Label 2-4026-1.1-c1-0-85
Degree $2$
Conductor $4026$
Sign $-1$
Analytic cond. $32.1477$
Root an. cond. $5.66990$
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  + 2-s − 3-s + 4-s + 1.40·5-s − 6-s − 3.03·7-s + 8-s + 9-s + 1.40·10-s − 11-s − 12-s + 3.73·13-s − 3.03·14-s − 1.40·15-s + 16-s − 7.21·17-s + 18-s + 6.74·19-s + 1.40·20-s + 3.03·21-s − 22-s − 5.31·23-s − 24-s − 3.01·25-s + 3.73·26-s − 27-s − 3.03·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.629·5-s − 0.408·6-s − 1.14·7-s + 0.353·8-s + 0.333·9-s + 0.445·10-s − 0.301·11-s − 0.288·12-s + 1.03·13-s − 0.810·14-s − 0.363·15-s + 0.250·16-s − 1.74·17-s + 0.235·18-s + 1.54·19-s + 0.314·20-s + 0.662·21-s − 0.213·22-s − 1.10·23-s − 0.204·24-s − 0.603·25-s + 0.732·26-s − 0.192·27-s − 0.573·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4026\)    =    \(2 \cdot 3 \cdot 11 \cdot 61\)
Sign: $-1$
Analytic conductor: \(32.1477\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4026,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 + T \)
11 \( 1 + T \)
61 \( 1 - T \)
good5 \( 1 - 1.40T + 5T^{2} \)
7 \( 1 + 3.03T + 7T^{2} \)
13 \( 1 - 3.73T + 13T^{2} \)
17 \( 1 + 7.21T + 17T^{2} \)
19 \( 1 - 6.74T + 19T^{2} \)
23 \( 1 + 5.31T + 23T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + 0.0646T + 31T^{2} \)
37 \( 1 - 1.89T + 37T^{2} \)
41 \( 1 + 7.30T + 41T^{2} \)
43 \( 1 - 9.52T + 43T^{2} \)
47 \( 1 - 4.75T + 47T^{2} \)
53 \( 1 + 7.05T + 53T^{2} \)
59 \( 1 - 2.27T + 59T^{2} \)
67 \( 1 + 8.39T + 67T^{2} \)
71 \( 1 + 2.31T + 71T^{2} \)
73 \( 1 - 6.68T + 73T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 - 2.10T + 83T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−7.85240165987814252996831967017, −7.09121341464750194927906182823, −6.32390770847037434344760124949, −5.89517625600045649030421348086, −5.30411238458266671259298008227, −4.17700260136035382122556047690, −3.58372307040851870079948599879, −2.54865183163965335659207529303, −1.58346327313316676378374994541, 0, 1.58346327313316676378374994541, 2.54865183163965335659207529303, 3.58372307040851870079948599879, 4.17700260136035382122556047690, 5.30411238458266671259298008227, 5.89517625600045649030421348086, 6.32390770847037434344760124949, 7.09121341464750194927906182823, 7.85240165987814252996831967017

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