Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 13 \cdot 103 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 3.32·5-s + 6-s − 1.38·7-s − 8-s + 9-s − 3.32·10-s − 2.60·11-s − 12-s + 13-s + 1.38·14-s − 3.32·15-s + 16-s − 1.65·17-s − 18-s − 2.38·19-s + 3.32·20-s + 1.38·21-s + 2.60·22-s + 0.437·23-s + 24-s + 6.08·25-s − 26-s − 27-s − 1.38·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.48·5-s + 0.408·6-s − 0.523·7-s − 0.353·8-s + 0.333·9-s − 1.05·10-s − 0.786·11-s − 0.288·12-s + 0.277·13-s + 0.370·14-s − 0.859·15-s + 0.250·16-s − 0.400·17-s − 0.235·18-s − 0.547·19-s + 0.744·20-s + 0.302·21-s + 0.556·22-s + 0.0912·23-s + 0.204·24-s + 1.21·25-s − 0.196·26-s − 0.192·27-s − 0.261·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8034,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;13,\;103\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;13,\;103\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 - T \)
good5 \( 1 - 3.32T + 5T^{2} \)
7 \( 1 + 1.38T + 7T^{2} \)
11 \( 1 + 2.60T + 11T^{2} \)
17 \( 1 + 1.65T + 17T^{2} \)
19 \( 1 + 2.38T + 19T^{2} \)
23 \( 1 - 0.437T + 23T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 + 5.09T + 31T^{2} \)
37 \( 1 - 6.25T + 37T^{2} \)
41 \( 1 - 2.88T + 41T^{2} \)
43 \( 1 - 0.927T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 - 8.01T + 53T^{2} \)
59 \( 1 - 3.33T + 59T^{2} \)
61 \( 1 + 1.16T + 61T^{2} \)
67 \( 1 - 8.02T + 67T^{2} \)
71 \( 1 - 5.77T + 71T^{2} \)
73 \( 1 + 5.91T + 73T^{2} \)
79 \( 1 - 5.61T + 79T^{2} \)
83 \( 1 + 7.43T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 + 1.80T + 97T^{2} \)
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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

−7.45673782773044960060209545053, −6.51481796271875646072701921419, −6.34539520914179746819526794426, −5.51843552442086294700054197826, −4.99505031590896607439905379812, −3.85657427857645547740818874451, −2.73361930513164286575816266526, −2.13847159063878761120244822571, −1.21204324084739057014127770493, 0, 1.21204324084739057014127770493, 2.13847159063878761120244822571, 2.73361930513164286575816266526, 3.85657427857645547740818874451, 4.99505031590896607439905379812, 5.51843552442086294700054197826, 6.34539520914179746819526794426, 6.51481796271875646072701921419, 7.45673782773044960060209545053

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