Properties

Degree 2
Conductor $ 2 \cdot 7^{2} \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 + 8-s + 9-s − 2·12-s − 4·13-s + 16-s − 6·17-s + 18-s − 2·24-s − 5·25-s − 4·26-s + 4·27-s + 6·29-s − 4·31-s + 32-s − 6·34-s + 36-s − 2·37-s + 8·39-s + 6·41-s + 8·43-s + 12·47-s − 2·48-s − 5·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.577·12-s − 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.408·24-s − 25-s − 0.784·26-s + 0.769·27-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 + 1.21·43-s + 1.75·47-s − 0.288·48-s − 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(35378\)    =    \(2 \cdot 7^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{35378} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 35378,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.030824817$
$L(\frac12)$  $\approx$  $1.030824817$
$L(\frac{3}{2})$   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 \( 1 \)
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

−14.99728502401873, −14.25983054662366, −13.94531513785829, −13.35276887883669, −12.56731102069754, −12.37388464531861, −11.87873551398178, −11.26962720487718, −10.81783723073702, −10.49521792945982, −9.664486576660142, −9.156955853739614, −8.455785059232210, −7.593989850498145, −7.226035651822059, −6.498461763115004, −6.106204737633401, −5.506725255735893, −4.963388069368697, −4.379223996486467, −3.957921222641953, −2.770787167079069, −2.428566160315683, −1.438665858952500, −0.3562927379342628, 0.3562927379342628, 1.438665858952500, 2.428566160315683, 2.770787167079069, 3.957921222641953, 4.379223996486467, 4.963388069368697, 5.506725255735893, 6.106204737633401, 6.498461763115004, 7.226035651822059, 7.593989850498145, 8.455785059232210, 9.156955853739614, 9.664486576660142, 10.49521792945982, 10.81783723073702, 11.26962720487718, 11.87873551398178, 12.37388464531861, 12.56731102069754, 13.35276887883669, 13.94531513785829, 14.25983054662366, 14.99728502401873

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