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  + 1.15·3-s + 3.94·5-s − 1.66·9-s + 1.12·11-s + 3.54·13-s + 4.55·15-s + 3.41·17-s − 1.28·19-s − 3.01·23-s + 10.5·25-s − 5.39·27-s + 7.53·29-s − 0.123·31-s + 1.30·33-s + 0.924·37-s + 4.09·39-s − 41-s − 3.99·43-s − 6.56·45-s + 1.47·47-s + 3.94·51-s + 8.78·53-s + 4.44·55-s − 1.48·57-s − 3.03·59-s + 12.2·61-s + 13.9·65-s + ⋯
L(s)  = 1  + 0.667·3-s + 1.76·5-s − 0.554·9-s + 0.339·11-s + 0.982·13-s + 1.17·15-s + 0.828·17-s − 0.294·19-s − 0.628·23-s + 2.11·25-s − 1.03·27-s + 1.39·29-s − 0.0221·31-s + 0.226·33-s + 0.151·37-s + 0.655·39-s − 0.156·41-s − 0.608·43-s − 0.979·45-s + 0.214·47-s + 0.552·51-s + 1.20·53-s + 0.598·55-s − 0.196·57-s − 0.394·59-s + 1.56·61-s + 1.73·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.145035370$
$L(\frac12)$  $\approx$  $4.145035370$
$L(\frac{3}{2})$   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 \( 1 - 1.15T + 3T^{2} \)
5 \( 1 - 3.94T + 5T^{2} \)
11 \( 1 - 1.12T + 11T^{2} \)
13 \( 1 - 3.54T + 13T^{2} \)
17 \( 1 - 3.41T + 17T^{2} \)
19 \( 1 + 1.28T + 19T^{2} \)
23 \( 1 + 3.01T + 23T^{2} \)
29 \( 1 - 7.53T + 29T^{2} \)
31 \( 1 + 0.123T + 31T^{2} \)
37 \( 1 - 0.924T + 37T^{2} \)
43 \( 1 + 3.99T + 43T^{2} \)
47 \( 1 - 1.47T + 47T^{2} \)
53 \( 1 - 8.78T + 53T^{2} \)
59 \( 1 + 3.03T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 2.72T + 67T^{2} \)
71 \( 1 - 5.72T + 71T^{2} \)
73 \( 1 + 7.47T + 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + 0.778T + 89T^{2} \)
97 \( 1 + 5.90T + 97T^{2} \)
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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

−8.137558977996208587253620876549, −6.98203476339354612935668621157, −6.35088092734798360237306710911, −5.77232971721049690561907483861, −5.29603490905693235134973766424, −4.20287503654204961892048048372, −3.30747880113442037400479912750, −2.61115899985710963591489876056, −1.87548504442919450765416480009, −1.03539691291421741204186305550, 1.03539691291421741204186305550, 1.87548504442919450765416480009, 2.61115899985710963591489876056, 3.30747880113442037400479912750, 4.20287503654204961892048048372, 5.29603490905693235134973766424, 5.77232971721049690561907483861, 6.35088092734798360237306710911, 6.98203476339354612935668621157, 8.137558977996208587253620876549

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