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  + 0.538·3-s + 0.810·5-s − 2.70·9-s + 4.92·11-s + 0.0654·13-s + 0.436·15-s − 1.24·17-s + 3.82·19-s − 1.71·23-s − 4.34·25-s − 3.07·27-s + 3.89·29-s + 3.19·31-s + 2.65·33-s − 0.0930·37-s + 0.0352·39-s − 41-s + 10.5·43-s − 2.19·45-s − 0.959·47-s − 0.668·51-s − 2.72·53-s + 3.99·55-s + 2.06·57-s + 11.1·59-s − 10.8·61-s + 0.0530·65-s + ⋯
L(s)  = 1  + 0.311·3-s + 0.362·5-s − 0.903·9-s + 1.48·11-s + 0.0181·13-s + 0.112·15-s − 0.300·17-s + 0.877·19-s − 0.358·23-s − 0.868·25-s − 0.592·27-s + 0.724·29-s + 0.574·31-s + 0.462·33-s − 0.0152·37-s + 0.00564·39-s − 0.156·41-s + 1.61·43-s − 0.327·45-s − 0.139·47-s − 0.0935·51-s − 0.374·53-s + 0.538·55-s + 0.272·57-s + 1.45·59-s − 1.38·61-s + 0.00657·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.534744122$
$L(\frac12)$  $\approx$  $2.534744122$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 - 0.538T + 3T^{2} \)
5 \( 1 - 0.810T + 5T^{2} \)
11 \( 1 - 4.92T + 11T^{2} \)
13 \( 1 - 0.0654T + 13T^{2} \)
17 \( 1 + 1.24T + 17T^{2} \)
19 \( 1 - 3.82T + 19T^{2} \)
23 \( 1 + 1.71T + 23T^{2} \)
29 \( 1 - 3.89T + 29T^{2} \)
31 \( 1 - 3.19T + 31T^{2} \)
37 \( 1 + 0.0930T + 37T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 0.959T + 47T^{2} \)
53 \( 1 + 2.72T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 5.60T + 71T^{2} \)
73 \( 1 - 2.76T + 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 - 9.99T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 - 6.31T + 97T^{2} \)
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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

−7.87358120838837014386997985431, −7.15905952752413323590543393472, −6.27381204636410871669909957608, −5.96318255424222691908383277801, −5.05748958728778189380939293233, −4.17597334799446651045481306338, −3.50474589688787021113056034616, −2.67373014487682916019203955527, −1.82194441739813767944674370911, −0.78884935137199995048637517826, 0.78884935137199995048637517826, 1.82194441739813767944674370911, 2.67373014487682916019203955527, 3.50474589688787021113056034616, 4.17597334799446651045481306338, 5.05748958728778189380939293233, 5.96318255424222691908383277801, 6.27381204636410871669909957608, 7.15905952752413323590543393472, 7.87358120838837014386997985431

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