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 + 6-s + 8-s + 9-s + 2·11-s + 12-s + 13-s + 16-s − 2·17-s + 18-s + 8·19-s + 2·22-s + 24-s − 5·25-s + 26-s + 27-s − 2·29-s + 10·31-s + 32-s + 2·33-s − 2·34-s + 36-s + 8·38-s + 39-s + 2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.83·19-s + 0.426·22-s + 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s − 0.371·29-s + 1.79·31-s + 0.176·32-s + 0.348·33-s − 0.342·34-s + 1/6·36-s + 1.29·38-s + 0.160·39-s + 0.312·41-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  =  \((2,\ 8034,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.771644914\)
\(L(\frac12)\)  \(\approx\)  \(4.771644914\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.80133481495295843322819522442, −7.08745627395774677014389752993, −6.43696932264732667314272969259, −5.73167100334378367840725926975, −4.94120568059144299125201836565, −4.20734984483792387552922138060, −3.51193588232521867230999066289, −2.86600164880482890665807550991, −1.92860122178507931178977174687, −0.994033076292022934540922513142, 0.994033076292022934540922513142, 1.92860122178507931178977174687, 2.86600164880482890665807550991, 3.51193588232521867230999066289, 4.20734984483792387552922138060, 4.94120568059144299125201836565, 5.73167100334378367840725926975, 6.43696932264732667314272969259, 7.08745627395774677014389752993, 7.80133481495295843322819522442

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