Properties

Label 2-4033-1.1-c1-0-63
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  − 2.76·2-s − 1.75·3-s + 5.63·4-s − 3.34·5-s + 4.83·6-s − 0.176·7-s − 10.0·8-s + 0.0666·9-s + 9.25·10-s − 3.59·11-s − 9.87·12-s − 2.30·13-s + 0.487·14-s + 5.86·15-s + 16.4·16-s − 6.59·17-s − 0.184·18-s − 1.28·19-s − 18.8·20-s + 0.308·21-s + 9.92·22-s − 0.795·23-s + 17.5·24-s + 6.21·25-s + 6.36·26-s + 5.13·27-s − 0.994·28-s + ⋯
L(s)  = 1  − 1.95·2-s − 1.01·3-s + 2.81·4-s − 1.49·5-s + 1.97·6-s − 0.0666·7-s − 3.55·8-s + 0.0222·9-s + 2.92·10-s − 1.08·11-s − 2.84·12-s − 0.639·13-s + 0.130·14-s + 1.51·15-s + 4.12·16-s − 1.59·17-s − 0.0433·18-s − 0.294·19-s − 4.22·20-s + 0.0674·21-s + 2.11·22-s − 0.165·23-s + 3.59·24-s + 1.24·25-s + 1.24·26-s + 0.988·27-s − 0.187·28-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 \( 1 + 2.76T + 2T^{2} \)
3 \( 1 + 1.75T + 3T^{2} \)
5 \( 1 + 3.34T + 5T^{2} \)
7 \( 1 + 0.176T + 7T^{2} \)
11 \( 1 + 3.59T + 11T^{2} \)
13 \( 1 + 2.30T + 13T^{2} \)
17 \( 1 + 6.59T + 17T^{2} \)
19 \( 1 + 1.28T + 19T^{2} \)
23 \( 1 + 0.795T + 23T^{2} \)
29 \( 1 + 1.57T + 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
41 \( 1 - 2.22T + 41T^{2} \)
43 \( 1 - 4.85T + 43T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 - 1.00T + 53T^{2} \)
59 \( 1 - 5.13T + 59T^{2} \)
61 \( 1 + 0.373T + 61T^{2} \)
67 \( 1 - 5.08T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 0.467T + 79T^{2} \)
83 \( 1 + 2.93T + 83T^{2} \)
89 \( 1 - 6.61T + 89T^{2} \)
97 \( 1 + 3.26T + 97T^{2} \)
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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

−8.176492284163253023650853395823, −7.50950649770580304337702090864, −6.88477202385376340024289586344, −6.27577652106997140886644989720, −5.28564448767457634542597190826, −4.29859089429621871695717073224, −2.98813759689960149334111648204, −2.22559709171350695499106888735, −0.63451977981377217105337952481, 0, 0.63451977981377217105337952481, 2.22559709171350695499106888735, 2.98813759689960149334111648204, 4.29859089429621871695717073224, 5.28564448767457634542597190826, 6.27577652106997140886644989720, 6.88477202385376340024289586344, 7.50950649770580304337702090864, 8.176492284163253023650853395823

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