Properties

Label 2-4022-1.1-c1-0-123
Degree $2$
Conductor $4022$
Sign $1$
Analytic cond. $32.1158$
Root an. cond. $5.66708$
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 + 1.87·3-s + 4-s + 4.35·5-s + 1.87·6-s + 1.06·7-s + 8-s + 0.503·9-s + 4.35·10-s − 1.51·11-s + 1.87·12-s − 0.975·13-s + 1.06·14-s + 8.15·15-s + 16-s − 1.23·17-s + 0.503·18-s + 0.425·19-s + 4.35·20-s + 1.98·21-s − 1.51·22-s + 2.85·23-s + 1.87·24-s + 13.9·25-s − 0.975·26-s − 4.67·27-s + 1.06·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.08·3-s + 0.5·4-s + 1.94·5-s + 0.764·6-s + 0.401·7-s + 0.353·8-s + 0.167·9-s + 1.37·10-s − 0.457·11-s + 0.540·12-s − 0.270·13-s + 0.283·14-s + 2.10·15-s + 0.250·16-s − 0.299·17-s + 0.118·18-s + 0.0975·19-s + 0.974·20-s + 0.433·21-s − 0.323·22-s + 0.596·23-s + 0.382·24-s + 2.79·25-s − 0.191·26-s − 0.899·27-s + 0.200·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4022\)    =    \(2 \cdot 2011\)
Sign: $1$
Analytic conductor: \(32.1158\)
Root analytic conductor: \(5.66708\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4022,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.408457180\)
\(L(\frac12)\) \(\approx\) \(6.408457180\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
2011 \( 1 + T \)
good3 \( 1 - 1.87T + 3T^{2} \)
5 \( 1 - 4.35T + 5T^{2} \)
7 \( 1 - 1.06T + 7T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 + 0.975T + 13T^{2} \)
17 \( 1 + 1.23T + 17T^{2} \)
19 \( 1 - 0.425T + 19T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 + 5.39T + 31T^{2} \)
37 \( 1 + 2.12T + 37T^{2} \)
41 \( 1 - 0.565T + 41T^{2} \)
43 \( 1 + 9.17T + 43T^{2} \)
47 \( 1 - 1.18T + 47T^{2} \)
53 \( 1 - 7.03T + 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 + 0.639T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 + 0.812T + 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 - 4.87T + 97T^{2} \)
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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

−8.596858773420788518909672291034, −7.71091057942883469101966276102, −6.82424447434248592065309518077, −6.19299182945931032231646430468, −5.23820760180440596196453442726, −4.99213646333086789022468038770, −3.65611436512710250880409285315, −2.73755104476955973994086262492, −2.27373885281386135201355335287, −1.45467177561814277440051487522, 1.45467177561814277440051487522, 2.27373885281386135201355335287, 2.73755104476955973994086262492, 3.65611436512710250880409285315, 4.99213646333086789022468038770, 5.23820760180440596196453442726, 6.19299182945931032231646430468, 6.82424447434248592065309518077, 7.71091057942883469101966276102, 8.596858773420788518909672291034

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