Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 19^{2} $
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 + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s + 2·12-s + 4·13-s + 14-s + 16-s + 6·17-s + 18-s + 2·21-s + 2·24-s − 5·25-s + 4·26-s − 4·27-s + 28-s + 6·29-s + 4·31-s + 32-s + 6·34-s + 36-s − 2·37-s + 8·39-s − 6·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.436·21-s + 0.408·24-s − 25-s + 0.784·26-s − 0.769·27-s + 0.188·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-s + 1.28·39-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5054\)    =    \(2 \cdot 7 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5054} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5054,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.473593818$
$L(\frac12)$  $\approx$  $5.473593818$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
19 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.196197307405668368216926690290, −7.72200516209190044787449977815, −6.77298005536370046551425088692, −5.97907088754740589421145166260, −5.31155422432102619146983664079, −4.35906178911561015019581069777, −3.55246093072316681260396868353, −3.09738642858536778661614069445, −2.11458272020859388026266946767, −1.19460132813703026742735308794, 1.19460132813703026742735308794, 2.11458272020859388026266946767, 3.09738642858536778661614069445, 3.55246093072316681260396868353, 4.35906178911561015019581069777, 5.31155422432102619146983664079, 5.97907088754740589421145166260, 6.77298005536370046551425088692, 7.72200516209190044787449977815, 8.196197307405668368216926690290

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