Properties

Label 2-8036-1.1-c1-0-134
Degree $2$
Conductor $8036$
Sign $1$
Analytic cond. $64.1677$
Root an. cond. $8.01047$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 2·9-s − 5·11-s − 2·13-s + 15-s − 3·17-s − 3·19-s − 23-s − 4·25-s + 5·27-s − 10·29-s − 11·31-s + 5·33-s + 7·37-s + 2·39-s − 41-s + 8·43-s + 2·45-s − 7·47-s + 3·51-s − 11·53-s + 5·55-s + 3·57-s − 7·59-s − 61-s + 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.50·11-s − 0.554·13-s + 0.258·15-s − 0.727·17-s − 0.688·19-s − 0.208·23-s − 4/5·25-s + 0.962·27-s − 1.85·29-s − 1.97·31-s + 0.870·33-s + 1.15·37-s + 0.320·39-s − 0.156·41-s + 1.21·43-s + 0.298·45-s − 1.02·47-s + 0.420·51-s − 1.51·53-s + 0.674·55-s + 0.397·57-s − 0.911·59-s − 0.128·61-s + 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8036\)    =    \(2^{2} \cdot 7^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(64.1677\)
Root analytic conductor: \(8.01047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8036} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 8036,\ (\ :1/2),\ 1)\)

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)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.37452149415854490759475438502, −6.27277380267903664181483316614, −5.73476017033492620342590931613, −5.12116698583109715756592944231, −4.40217961167974444120615080952, −3.53152374673478715820591689153, −2.60966020994397495341144354733, −1.89324731327228348804965533350, 0, 0, 1.89324731327228348804965533350, 2.60966020994397495341144354733, 3.53152374673478715820591689153, 4.40217961167974444120615080952, 5.12116698583109715756592944231, 5.73476017033492620342590931613, 6.27277380267903664181483316614, 7.37452149415854490759475438502

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