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Properties

Label 2-4033-1.1-c1-0-214
Degree $2$
Conductor $4033$
Sign $-1$
Analytic cond. $32.2036$
Root an. cond. $5.67482$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s + 3·8-s − 3·9-s − 2·10-s + 2·11-s − 2·13-s − 16-s + 6·17-s + 3·18-s − 4·19-s − 2·20-s − 2·22-s − 4·23-s − 25-s + 2·26-s − 6·29-s + 10·31-s − 5·32-s − 6·34-s + 3·36-s − 37-s + 4·38-s + 6·40-s + 10·41-s − 10·43-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 9-s − 0.632·10-s + 0.603·11-s − 0.554·13-s − 1/4·16-s + 1.45·17-s + 0.707·18-s − 0.917·19-s − 0.447·20-s − 0.426·22-s − 0.834·23-s − 1/5·25-s + 0.392·26-s − 1.11·29-s + 1.79·31-s − 0.883·32-s − 1.02·34-s + 1/2·36-s − 0.164·37-s + 0.648·38-s + 0.948·40-s + 1.56·41-s − 1.52·43-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: yes
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 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.035487168142738731697918005286, −7.76002166302831371280675456665, −6.49028946934905162159967299502, −5.91862685006845857213491940161, −5.17915807181459376282104315557, −4.32382297931770495090952772133, −3.33565813799740588466353537804, −2.24518209799865911376722737131, −1.31506648185206171078630563411, 0, 1.31506648185206171078630563411, 2.24518209799865911376722737131, 3.33565813799740588466353537804, 4.32382297931770495090952772133, 5.17915807181459376282104315557, 5.91862685006845857213491940161, 6.49028946934905162159967299502, 7.76002166302831371280675456665, 8.035487168142738731697918005286

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