Properties

Degree 2
Conductor $ 2^{2} \cdot 7^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.42·3-s + 2.35·5-s + 8.72·9-s − 0.142·11-s − 2.89·13-s − 8.07·15-s − 1.39·17-s − 7.23·19-s − 4.70·23-s + 0.557·25-s − 19.5·27-s + 2.50·29-s − 2.29·31-s + 0.489·33-s − 8.95·37-s + 9.89·39-s − 41-s + 10.0·43-s + 20.5·45-s − 0.577·47-s + 4.77·51-s − 2.67·53-s − 0.336·55-s + 24.7·57-s − 5.00·59-s + 10.1·61-s − 6.81·65-s + ⋯
L(s)  = 1  − 1.97·3-s + 1.05·5-s + 2.90·9-s − 0.0430·11-s − 0.801·13-s − 2.08·15-s − 0.338·17-s − 1.65·19-s − 0.980·23-s + 0.111·25-s − 3.77·27-s + 0.464·29-s − 0.412·31-s + 0.0851·33-s − 1.47·37-s + 1.58·39-s − 0.156·41-s + 1.53·43-s + 3.06·45-s − 0.0842·47-s + 0.668·51-s − 0.367·53-s − 0.0454·55-s + 3.28·57-s − 0.651·59-s + 1.30·61-s − 0.845·65-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

\( d \)  =  \(2\)
\( N \)  =  \(8036\)    =    \(2^{2} \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6886208915$
$L(\frac12)$  $\approx$  $0.6886208915$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 + 3.42T + 3T^{2} \)
5 \( 1 - 2.35T + 5T^{2} \)
11 \( 1 + 0.142T + 11T^{2} \)
13 \( 1 + 2.89T + 13T^{2} \)
17 \( 1 + 1.39T + 17T^{2} \)
19 \( 1 + 7.23T + 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 - 2.50T + 29T^{2} \)
31 \( 1 + 2.29T + 31T^{2} \)
37 \( 1 + 8.95T + 37T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + 0.577T + 47T^{2} \)
53 \( 1 + 2.67T + 53T^{2} \)
59 \( 1 + 5.00T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 4.81T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + 2.06T + 73T^{2} \)
79 \( 1 - 2.74T + 79T^{2} \)
83 \( 1 - 8.58T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 4.73T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.49955940169038259391773342148, −6.89226135818552444570843575988, −6.19605243961153561323860038891, −5.92015099547153351628652550831, −5.13234733218789894804467953006, −4.59046687822490918929420048862, −3.85538207472432115806771460970, −2.23933095695166641313976802638, −1.73079340007142669518234532208, −0.44737255001197887266123154107, 0.44737255001197887266123154107, 1.73079340007142669518234532208, 2.23933095695166641313976802638, 3.85538207472432115806771460970, 4.59046687822490918929420048862, 5.13234733218789894804467953006, 5.92015099547153351628652550831, 6.19605243961153561323860038891, 6.89226135818552444570843575988, 7.49955940169038259391773342148

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