Properties

Label 2-4026-1.1-c1-0-11
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 3.69·5-s − 6-s + 3.64·7-s − 8-s + 9-s + 3.69·10-s + 11-s + 12-s − 6.50·13-s − 3.64·14-s − 3.69·15-s + 16-s − 6.91·17-s − 18-s + 2.83·19-s − 3.69·20-s + 3.64·21-s − 22-s − 5.46·23-s − 24-s + 8.68·25-s + 6.50·26-s + 27-s + 3.64·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.65·5-s − 0.408·6-s + 1.37·7-s − 0.353·8-s + 0.333·9-s + 1.16·10-s + 0.301·11-s + 0.288·12-s − 1.80·13-s − 0.974·14-s − 0.955·15-s + 0.250·16-s − 1.67·17-s − 0.235·18-s + 0.650·19-s − 0.827·20-s + 0.795·21-s − 0.213·22-s − 1.13·23-s − 0.204·24-s + 1.73·25-s + 1.27·26-s + 0.192·27-s + 0.689·28-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: \(0\)
Selberg data: \((2,\ 4026,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.059568366\)
\(L(\frac12)\) \(\approx\) \(1.059568366\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.69T + 5T^{2} \)
7 \( 1 - 3.64T + 7T^{2} \)
13 \( 1 + 6.50T + 13T^{2} \)
17 \( 1 + 6.91T + 17T^{2} \)
19 \( 1 - 2.83T + 19T^{2} \)
23 \( 1 + 5.46T + 23T^{2} \)
29 \( 1 + 2.39T + 29T^{2} \)
31 \( 1 - 3.76T + 31T^{2} \)
37 \( 1 - 1.16T + 37T^{2} \)
41 \( 1 - 4.51T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 - 7.72T + 47T^{2} \)
53 \( 1 + 5.76T + 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
67 \( 1 - 0.969T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 1.00T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−8.283033367141813561557200905001, −7.74301570498097947708359511939, −7.43369580754750904951097216781, −6.67226574170522127061726132344, −5.27726282282051648349248900820, −4.38263031799625901022399225693, −4.06012085153734909625909884947, −2.70135392310973632177050323602, −2.03052996581321614699337247080, −0.62017909927224501615766842410, 0.62017909927224501615766842410, 2.03052996581321614699337247080, 2.70135392310973632177050323602, 4.06012085153734909625909884947, 4.38263031799625901022399225693, 5.27726282282051648349248900820, 6.67226574170522127061726132344, 7.43369580754750904951097216781, 7.74301570498097947708359511939, 8.283033367141813561557200905001

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