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  − 1.91·3-s + 1.95·5-s + 0.651·9-s + 0.643·11-s − 3.67·13-s − 3.74·15-s − 4.14·17-s − 0.715·19-s − 6.03·23-s − 1.16·25-s + 4.48·27-s − 8.58·29-s + 6.28·31-s − 1.22·33-s + 5.47·37-s + 7.02·39-s + 41-s − 10.5·43-s + 1.27·45-s − 9.33·47-s + 7.92·51-s − 0.928·53-s + 1.26·55-s + 1.36·57-s + 3.27·59-s + 3.37·61-s − 7.19·65-s + ⋯
L(s)  = 1  − 1.10·3-s + 0.875·5-s + 0.217·9-s + 0.194·11-s − 1.01·13-s − 0.966·15-s − 1.00·17-s − 0.164·19-s − 1.25·23-s − 0.233·25-s + 0.863·27-s − 1.59·29-s + 1.12·31-s − 0.214·33-s + 0.900·37-s + 1.12·39-s + 0.156·41-s − 1.61·43-s + 0.190·45-s − 1.36·47-s + 1.10·51-s − 0.127·53-s + 0.169·55-s + 0.181·57-s + 0.426·59-s + 0.431·61-s − 0.892·65-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

\(L(1)\) \(\approx\) \(0.8672186239\)
\(L(\frac12)\) \(\approx\) \(0.8672186239\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.91T + 3T^{2} \)
5 \( 1 - 1.95T + 5T^{2} \)
11 \( 1 - 0.643T + 11T^{2} \)
13 \( 1 + 3.67T + 13T^{2} \)
17 \( 1 + 4.14T + 17T^{2} \)
19 \( 1 + 0.715T + 19T^{2} \)
23 \( 1 + 6.03T + 23T^{2} \)
29 \( 1 + 8.58T + 29T^{2} \)
31 \( 1 - 6.28T + 31T^{2} \)
37 \( 1 - 5.47T + 37T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 9.33T + 47T^{2} \)
53 \( 1 + 0.928T + 53T^{2} \)
59 \( 1 - 3.27T + 59T^{2} \)
61 \( 1 - 3.37T + 61T^{2} \)
67 \( 1 - 13.9T + 67T^{2} \)
71 \( 1 + 0.555T + 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 - 3.09T + 79T^{2} \)
83 \( 1 + 6.57T + 83T^{2} \)
89 \( 1 - 16.3T + 89T^{2} \)
97 \( 1 - 18.8T + 97T^{2} \)
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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

−7.82054408723683628641023241735, −6.80432514242118125729083269254, −6.40177518735444524308032162584, −5.79811922518367289434217594437, −5.11768949327371766481646578341, −4.57067360569974408890440679672, −3.60177585659674993338186756100, −2.38678231195116075863502313380, −1.85501560562082013339845883393, −0.46694016863208501644468958117, 0.46694016863208501644468958117, 1.85501560562082013339845883393, 2.38678231195116075863502313380, 3.60177585659674993338186756100, 4.57067360569974408890440679672, 5.11768949327371766481646578341, 5.79811922518367289434217594437, 6.40177518735444524308032162584, 6.80432514242118125729083269254, 7.82054408723683628641023241735

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