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  − 2.71·2-s + 2.16·3-s + 5.36·4-s + 5-s − 5.87·6-s − 4.59·7-s − 9.14·8-s + 1.67·9-s − 2.71·10-s − 1.85·11-s + 11.6·12-s − 0.473·13-s + 12.4·14-s + 2.16·15-s + 14.0·16-s + 5.22·17-s − 4.55·18-s + 6.43·19-s + 5.36·20-s − 9.93·21-s + 5.04·22-s − 6.10·23-s − 19.7·24-s + 25-s + 1.28·26-s − 2.85·27-s − 24.6·28-s + ⋯
L(s)  = 1  − 1.91·2-s + 1.24·3-s + 2.68·4-s + 0.447·5-s − 2.39·6-s − 1.73·7-s − 3.23·8-s + 0.559·9-s − 0.858·10-s − 0.560·11-s + 3.35·12-s − 0.131·13-s + 3.33·14-s + 0.558·15-s + 3.52·16-s + 1.26·17-s − 1.07·18-s + 1.47·19-s + 1.20·20-s − 2.16·21-s + 1.07·22-s − 1.27·23-s − 4.03·24-s + 0.200·25-s + 0.252·26-s − 0.550·27-s − 4.65·28-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  =  $(2,\ 8035,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9634183887$
$L(\frac12)$  $\approx$  $0.9634183887$
$L(\frac{3}{2})$   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 \( 1 + 2.71T + 2T^{2} \)
3 \( 1 - 2.16T + 3T^{2} \)
7 \( 1 + 4.59T + 7T^{2} \)
11 \( 1 + 1.85T + 11T^{2} \)
13 \( 1 + 0.473T + 13T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
19 \( 1 - 6.43T + 19T^{2} \)
23 \( 1 + 6.10T + 23T^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 + 3.89T + 31T^{2} \)
37 \( 1 + 0.772T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 4.44T + 43T^{2} \)
47 \( 1 + 3.40T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 1.03T + 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 + 0.947T + 79T^{2} \)
83 \( 1 - 5.84T + 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 + 4.41T + 97T^{2} \)
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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

−8.147222451972457644602909199194, −7.23680431647939820349432088576, −6.98089734956009809677268455470, −6.02308286747205036655708575642, −5.47720986913545789001771966547, −3.60982407454104643636158385398, −3.06614222209034217908393127735, −2.63257990270336323643432604339, −1.69091390976363888905447239927, −0.59318857582758304773129768087, 0.59318857582758304773129768087, 1.69091390976363888905447239927, 2.63257990270336323643432604339, 3.06614222209034217908393127735, 3.60982407454104643636158385398, 5.47720986913545789001771966547, 6.02308286747205036655708575642, 6.98089734956009809677268455470, 7.23680431647939820349432088576, 8.147222451972457644602909199194

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