Properties

Label 2-4033-1.1-c1-0-286
Degree $2$
Conductor $4033$
Sign $-1$
Analytic cond. $32.2036$
Root an. cond. $5.67482$
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  − 2.60·2-s + 3.31·3-s + 4.76·4-s − 1.17·5-s − 8.61·6-s + 2.01·7-s − 7.19·8-s + 7.97·9-s + 3.04·10-s − 2.35·11-s + 15.7·12-s − 1.55·13-s − 5.24·14-s − 3.88·15-s + 9.17·16-s − 5.14·17-s − 20.7·18-s + 1.38·19-s − 5.58·20-s + 6.67·21-s + 6.13·22-s − 7.77·23-s − 23.8·24-s − 3.62·25-s + 4.04·26-s + 16.5·27-s + 9.60·28-s + ⋯
L(s)  = 1  − 1.83·2-s + 1.91·3-s + 2.38·4-s − 0.523·5-s − 3.51·6-s + 0.761·7-s − 2.54·8-s + 2.65·9-s + 0.963·10-s − 0.710·11-s + 4.55·12-s − 0.431·13-s − 1.40·14-s − 1.00·15-s + 2.29·16-s − 1.24·17-s − 4.89·18-s + 0.317·19-s − 1.24·20-s + 1.45·21-s + 1.30·22-s − 1.62·23-s − 4.86·24-s − 0.725·25-s + 0.792·26-s + 3.17·27-s + 1.81·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-1$
Analytic conductor: \(32.2036\)
Root analytic conductor: \(5.67482\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4033,\ (\ :1/2),\ -1)\)

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)$
bad37 \( 1 + T \)
109 \( 1 + T \)
good2 \( 1 + 2.60T + 2T^{2} \)
3 \( 1 - 3.31T + 3T^{2} \)
5 \( 1 + 1.17T + 5T^{2} \)
7 \( 1 - 2.01T + 7T^{2} \)
11 \( 1 + 2.35T + 11T^{2} \)
13 \( 1 + 1.55T + 13T^{2} \)
17 \( 1 + 5.14T + 17T^{2} \)
19 \( 1 - 1.38T + 19T^{2} \)
23 \( 1 + 7.77T + 23T^{2} \)
29 \( 1 - 0.340T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
41 \( 1 - 2.43T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + 8.86T + 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 - 2.65T + 59T^{2} \)
61 \( 1 - 4.90T + 61T^{2} \)
67 \( 1 + 2.27T + 67T^{2} \)
71 \( 1 + 1.52T + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 - 5.42T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 - 9.49T + 89T^{2} \)
97 \( 1 - 0.204T + 97T^{2} \)
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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

−8.140491200277348474738122400619, −7.86233162047528974564827462553, −7.20527065284169204439278138412, −6.45914657428413412444443240099, −4.90789808277856015784990441864, −3.90073449826927615774067715547, −2.99593205778721294313606581874, −2.04529931917132111712382546938, −1.74799055274528024228995111296, 0, 1.74799055274528024228995111296, 2.04529931917132111712382546938, 2.99593205778721294313606581874, 3.90073449826927615774067715547, 4.90789808277856015784990441864, 6.45914657428413412444443240099, 7.20527065284169204439278138412, 7.86233162047528974564827462553, 8.140491200277348474738122400619

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