Properties

Degree 2
Conductor $ 5 \cdot 1607 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.51·2-s − 1.73·3-s + 4.34·4-s + 5-s + 4.37·6-s − 5.14·7-s − 5.91·8-s + 0.0182·9-s − 2.51·10-s + 1.49·11-s − 7.55·12-s − 5.55·13-s + 12.9·14-s − 1.73·15-s + 6.19·16-s − 2.59·17-s − 0.0459·18-s + 4.66·19-s + 4.34·20-s + 8.94·21-s − 3.77·22-s + 1.25·23-s + 10.2·24-s + 25-s + 13.9·26-s + 5.18·27-s − 22.3·28-s + ⋯
L(s)  = 1  − 1.78·2-s − 1.00·3-s + 2.17·4-s + 0.447·5-s + 1.78·6-s − 1.94·7-s − 2.09·8-s + 0.00607·9-s − 0.796·10-s + 0.451·11-s − 2.17·12-s − 1.53·13-s + 3.46·14-s − 0.448·15-s + 1.54·16-s − 0.629·17-s − 0.0108·18-s + 1.07·19-s + 0.971·20-s + 1.95·21-s − 0.804·22-s + 0.261·23-s + 2.09·24-s + 0.200·25-s + 2.74·26-s + 0.996·27-s − 4.22·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8035\)    =    \(5 \cdot 1607\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8035} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8035,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.09761227946$
$L(\frac12)$  $\approx$  $0.09761227946$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;1607\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;1607\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 - T \)
1607 \( 1 + T \)
good2 \( 1 + 2.51T + 2T^{2} \)
3 \( 1 + 1.73T + 3T^{2} \)
7 \( 1 + 5.14T + 7T^{2} \)
11 \( 1 - 1.49T + 11T^{2} \)
13 \( 1 + 5.55T + 13T^{2} \)
17 \( 1 + 2.59T + 17T^{2} \)
19 \( 1 - 4.66T + 19T^{2} \)
23 \( 1 - 1.25T + 23T^{2} \)
29 \( 1 + 2.96T + 29T^{2} \)
31 \( 1 + 6.58T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 - 1.49T + 41T^{2} \)
43 \( 1 - 9.01T + 43T^{2} \)
47 \( 1 + 2.14T + 47T^{2} \)
53 \( 1 + 8.39T + 53T^{2} \)
59 \( 1 + 8.30T + 59T^{2} \)
61 \( 1 - 1.72T + 61T^{2} \)
67 \( 1 + 14.6T + 67T^{2} \)
71 \( 1 - 2.20T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 + 0.117T + 79T^{2} \)
83 \( 1 + 9.42T + 83T^{2} \)
89 \( 1 + 17.9T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−7.60360665842032933881162996533, −7.23914001767166430723945891167, −6.58054160268335898561790271862, −6.04795145269801839322682352215, −5.50028666648743034837852913329, −4.32050497892790989677007345570, −2.95132065012393120052070553449, −2.60575617389627600658946287035, −1.29978617178097686903323259465, −0.22656054843855900650895781130, 0.22656054843855900650895781130, 1.29978617178097686903323259465, 2.60575617389627600658946287035, 2.95132065012393120052070553449, 4.32050497892790989677007345570, 5.50028666648743034837852913329, 6.04795145269801839322682352215, 6.58054160268335898561790271862, 7.23914001767166430723945891167, 7.60360665842032933881162996533

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