Properties

Degree $2$
Conductor $8036$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.20·3-s − 0.960·5-s − 1.55·9-s − 4.42·11-s + 0.696·13-s − 1.15·15-s + 1.02·17-s − 7.03·19-s − 4.50·23-s − 4.07·25-s − 5.47·27-s + 8.50·29-s + 2.30·31-s − 5.30·33-s + 6.54·37-s + 0.836·39-s + 41-s + 6.37·43-s + 1.49·45-s + 0.286·47-s + 1.22·51-s + 6.03·53-s + 4.24·55-s − 8.44·57-s − 14.2·59-s + 5.32·61-s − 0.669·65-s + ⋯
L(s)  = 1  + 0.693·3-s − 0.429·5-s − 0.519·9-s − 1.33·11-s + 0.193·13-s − 0.297·15-s + 0.247·17-s − 1.61·19-s − 0.939·23-s − 0.815·25-s − 1.05·27-s + 1.57·29-s + 0.413·31-s − 0.924·33-s + 1.07·37-s + 0.133·39-s + 0.156·41-s + 0.971·43-s + 0.223·45-s + 0.0418·47-s + 0.171·51-s + 0.828·53-s + 0.572·55-s − 1.11·57-s − 1.85·59-s + 0.681·61-s − 0.0829·65-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: $1$
Motivic weight: \(1\)
Character: $\chi_{8036} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8036,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.433336309\)
\(L(\frac12)\) \(\approx\) \(1.433336309\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.20T + 3T^{2} \)
5 \( 1 + 0.960T + 5T^{2} \)
11 \( 1 + 4.42T + 11T^{2} \)
13 \( 1 - 0.696T + 13T^{2} \)
17 \( 1 - 1.02T + 17T^{2} \)
19 \( 1 + 7.03T + 19T^{2} \)
23 \( 1 + 4.50T + 23T^{2} \)
29 \( 1 - 8.50T + 29T^{2} \)
31 \( 1 - 2.30T + 31T^{2} \)
37 \( 1 - 6.54T + 37T^{2} \)
43 \( 1 - 6.37T + 43T^{2} \)
47 \( 1 - 0.286T + 47T^{2} \)
53 \( 1 - 6.03T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 - 5.32T + 61T^{2} \)
67 \( 1 - 0.778T + 67T^{2} \)
71 \( 1 + 0.249T + 71T^{2} \)
73 \( 1 - 6.06T + 73T^{2} \)
79 \( 1 - 1.79T + 79T^{2} \)
83 \( 1 - 0.273T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−7.943860225753268808158128656286, −7.46457968937693862209502494025, −6.26105517860311485585238616590, −5.95204711453796317340903935277, −4.89460303007958168132219232987, −4.23491866774568803246184281379, −3.44641672386074432748605770163, −2.58093579966122292100818640842, −2.12993301204563500281882057720, −0.53637482350666364807205846753, 0.53637482350666364807205846753, 2.12993301204563500281882057720, 2.58093579966122292100818640842, 3.44641672386074432748605770163, 4.23491866774568803246184281379, 4.89460303007958168132219232987, 5.95204711453796317340903935277, 6.26105517860311485585238616590, 7.46457968937693862209502494025, 7.943860225753268808158128656286

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