Properties

Degree 2
Conductor $ 7 \cdot 863 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.57·2-s − 0.937·3-s + 4.64·4-s − 3.87·5-s + 2.41·6-s + 7-s − 6.81·8-s − 2.12·9-s + 9.99·10-s + 4.77·11-s − 4.35·12-s − 5.45·13-s − 2.57·14-s + 3.63·15-s + 8.28·16-s − 2.70·17-s + 5.46·18-s − 0.0387·19-s − 18.0·20-s − 0.937·21-s − 12.2·22-s + 8.78·23-s + 6.39·24-s + 10.0·25-s + 14.0·26-s + 4.80·27-s + 4.64·28-s + ⋯
L(s)  = 1  − 1.82·2-s − 0.541·3-s + 2.32·4-s − 1.73·5-s + 0.986·6-s + 0.377·7-s − 2.41·8-s − 0.706·9-s + 3.16·10-s + 1.43·11-s − 1.25·12-s − 1.51·13-s − 0.688·14-s + 0.939·15-s + 2.07·16-s − 0.655·17-s + 1.28·18-s − 0.00888·19-s − 4.02·20-s − 0.204·21-s − 2.62·22-s + 1.83·23-s + 1.30·24-s + 2.00·25-s + 2.75·26-s + 0.924·27-s + 0.877·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6041\)    =    \(7 \cdot 863\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6041} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6041,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2229073223$
$L(\frac12)$  $\approx$  $0.2229073223$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;863\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;863\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 - T \)
863 \( 1 + T \)
good2 \( 1 + 2.57T + 2T^{2} \)
3 \( 1 + 0.937T + 3T^{2} \)
5 \( 1 + 3.87T + 5T^{2} \)
11 \( 1 - 4.77T + 11T^{2} \)
13 \( 1 + 5.45T + 13T^{2} \)
17 \( 1 + 2.70T + 17T^{2} \)
19 \( 1 + 0.0387T + 19T^{2} \)
23 \( 1 - 8.78T + 23T^{2} \)
29 \( 1 + 5.81T + 29T^{2} \)
31 \( 1 - 7.97T + 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 + 1.45T + 41T^{2} \)
43 \( 1 - 6.85T + 43T^{2} \)
47 \( 1 + 7.22T + 47T^{2} \)
53 \( 1 + 8.65T + 53T^{2} \)
59 \( 1 + 9.38T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 + 0.538T + 67T^{2} \)
71 \( 1 - 4.14T + 71T^{2} \)
73 \( 1 - 0.785T + 73T^{2} \)
79 \( 1 - 6.77T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−8.106619257246634243869509281660, −7.53709166378521402953944293113, −6.83775018286369531533617230354, −6.58468039127344334449287032654, −5.19733046460698369249967701671, −4.49174786151603865513107255301, −3.40356993382563071214824126383, −2.57925924905287333987249543142, −1.31786360756810388334900661516, −0.36648567336228768851790083608, 0.36648567336228768851790083608, 1.31786360756810388334900661516, 2.57925924905287333987249543142, 3.40356993382563071214824126383, 4.49174786151603865513107255301, 5.19733046460698369249967701671, 6.58468039127344334449287032654, 6.83775018286369531533617230354, 7.53709166378521402953944293113, 8.106619257246634243869509281660

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