Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 7^{2} \cdot 11^{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 − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s − 4·13-s + 16-s − 6·17-s + 18-s − 4·19-s + 6·23-s − 24-s − 5·25-s − 4·26-s − 27-s − 6·29-s − 8·31-s + 32-s − 6·34-s + 36-s − 10·37-s − 4·38-s + 4·39-s + 6·41-s − 8·43-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 − 0.288·12-s − 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s + 1.25·23-s − 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 1.64·37-s − 0.648·38-s + 0.640·39-s + 0.937·41-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(35574\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{35574} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 35574,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.090044023$
$L(\frac12)$  $\approx$  $1.090044023$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 \)
11 \( 1 \)
good5 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.08156663141700, −14.37131869315095, −13.91856737773756, −13.12935560708884, −12.83396328954375, −12.53463903946886, −11.57570079822976, −11.46315889703677, −10.79894693930279, −10.35450088116840, −9.675250670490676, −9.004043487745736, −8.591422558136058, −7.570044845724305, −7.180756845427999, −6.744621423956252, −6.069373750903660, −5.365086869220132, −5.053450758983110, −4.261144321729899, −3.869252143439276, −2.968958037556151, −2.127103415487960, −1.745934426318372, −0.3319756581567727, 0.3319756581567727, 1.745934426318372, 2.127103415487960, 2.968958037556151, 3.869252143439276, 4.261144321729899, 5.053450758983110, 5.365086869220132, 6.069373750903660, 6.744621423956252, 7.180756845427999, 7.570044845724305, 8.591422558136058, 9.004043487745736, 9.675250670490676, 10.35450088116840, 10.79894693930279, 11.46315889703677, 11.57570079822976, 12.53463903946886, 12.83396328954375, 13.12935560708884, 13.91856737773756, 14.37131869315095, 15.08156663141700

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