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 − 2.55·5-s + 6-s + 1.37·7-s − 8-s + 9-s + 2.55·10-s + 4.10·11-s − 12-s + 13-s − 1.37·14-s + 2.55·15-s + 16-s + 3.57·17-s − 18-s + 0.378·19-s − 2.55·20-s − 1.37·21-s − 4.10·22-s + 3.65·23-s + 24-s + 1.51·25-s − 26-s − 27-s + 1.37·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.14·5-s + 0.408·6-s + 0.520·7-s − 0.353·8-s + 0.333·9-s + 0.807·10-s + 1.23·11-s − 0.288·12-s + 0.277·13-s − 0.368·14-s + 0.659·15-s + 0.250·16-s + 0.866·17-s − 0.235·18-s + 0.0867·19-s − 0.570·20-s − 0.300·21-s − 0.875·22-s + 0.761·23-s + 0.204·24-s + 0.302·25-s − 0.196·26-s − 0.192·27-s + 0.260·28-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 \( 1 + 2.55T + 5T^{2} \)
7 \( 1 - 1.37T + 7T^{2} \)
11 \( 1 - 4.10T + 11T^{2} \)
17 \( 1 - 3.57T + 17T^{2} \)
19 \( 1 - 0.378T + 19T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 + 0.329T + 29T^{2} \)
31 \( 1 + 7.41T + 31T^{2} \)
37 \( 1 + 6.88T + 37T^{2} \)
41 \( 1 + 7.85T + 41T^{2} \)
43 \( 1 + 2.74T + 43T^{2} \)
47 \( 1 + 2.13T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 - 8.34T + 59T^{2} \)
61 \( 1 + 0.317T + 61T^{2} \)
67 \( 1 - 3.12T + 67T^{2} \)
71 \( 1 - 8.40T + 71T^{2} \)
73 \( 1 - 2.15T + 73T^{2} \)
79 \( 1 + 5.79T + 79T^{2} \)
83 \( 1 + 1.11T + 83T^{2} \)
89 \( 1 - 0.259T + 89T^{2} \)
97 \( 1 - 4.63T + 97T^{2} \)
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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

−7.43494703036450142459728500695, −6.99020955473435697421695943657, −6.30289281436524659617134852263, −5.38153153970625845817177478398, −4.73062427364344171482356053231, −3.69467298131860357973060529329, −3.38756499587384812261273501490, −1.81844847898486280794005411894, −1.12598799712367628695757540994, 0, 1.12598799712367628695757540994, 1.81844847898486280794005411894, 3.38756499587384812261273501490, 3.69467298131860357973060529329, 4.73062427364344171482356053231, 5.38153153970625845817177478398, 6.30289281436524659617134852263, 6.99020955473435697421695943657, 7.43494703036450142459728500695

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