Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 13 \cdot 103 $
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-s − 3-s + 4-s + 0.656·5-s − 6-s + 2.89·7-s + 8-s + 9-s + 0.656·10-s + 2.55·11-s − 12-s − 13-s + 2.89·14-s − 0.656·15-s + 16-s − 3.68·17-s + 18-s + 2.16·19-s + 0.656·20-s − 2.89·21-s + 2.55·22-s + 6.84·23-s − 24-s − 4.56·25-s − 26-s − 27-s + 2.89·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.293·5-s − 0.408·6-s + 1.09·7-s + 0.353·8-s + 0.333·9-s + 0.207·10-s + 0.769·11-s − 0.288·12-s − 0.277·13-s + 0.772·14-s − 0.169·15-s + 0.250·16-s − 0.892·17-s + 0.235·18-s + 0.496·19-s + 0.146·20-s − 0.631·21-s + 0.543·22-s + 1.42·23-s − 0.204·24-s − 0.913·25-s − 0.196·26-s − 0.192·27-s + 0.546·28-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  =  \(0\)
Selberg data  =  \((2,\ 8034,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.640294747\)
\(L(\frac12)\)  \(\approx\)  \(3.640294747\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
13 \( 1 + T \)
103 \( 1 + T \)
good5 \( 1 - 0.656T + 5T^{2} \)
7 \( 1 - 2.89T + 7T^{2} \)
11 \( 1 - 2.55T + 11T^{2} \)
17 \( 1 + 3.68T + 17T^{2} \)
19 \( 1 - 2.16T + 19T^{2} \)
23 \( 1 - 6.84T + 23T^{2} \)
29 \( 1 + 8.30T + 29T^{2} \)
31 \( 1 - 8.50T + 31T^{2} \)
37 \( 1 + 2.04T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 + 0.666T + 43T^{2} \)
47 \( 1 + 5.56T + 47T^{2} \)
53 \( 1 - 8.88T + 53T^{2} \)
59 \( 1 - 6.49T + 59T^{2} \)
61 \( 1 - 3.88T + 61T^{2} \)
67 \( 1 - 9.97T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + 2.34T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 + 0.205T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−7.68082618828548399089760098818, −6.92287206362771161250881989528, −6.45898057270695419176119543355, −5.50007279635129196715022326223, −5.16058256665019365169373934465, −4.34938650408398884787075424347, −3.79787784773860080829165896679, −2.60069319377897219104383310866, −1.81867388802421021464609924607, −0.919935281408565098577150507573, 0.919935281408565098577150507573, 1.81867388802421021464609924607, 2.60069319377897219104383310866, 3.79787784773860080829165896679, 4.34938650408398884787075424347, 5.16058256665019365169373934465, 5.50007279635129196715022326223, 6.45898057270695419176119543355, 6.92287206362771161250881989528, 7.68082618828548399089760098818

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