Properties

Degree 2
Conductor $ 2 \cdot 5^{2} \cdot 7 \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

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 − 2·19-s + 2·21-s − 2·24-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 + 2·38-s − 8·39-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 − 0.458·19-s + 0.436·21-s − 0.408·24-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 + 0.324·38-s − 1.28·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(42350\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{42350} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 42350,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
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

−14.92895295022713, −14.35415004838895, −14.18120843858814, −13.69057364271465, −12.66759575988296, −12.50760655863588, −11.84585823184490, −11.33422240767362, −10.48565238189661, −10.24437427913259, −9.616989158560158, −9.037420426536510, −8.744288985594174, −7.998637566918170, −7.658481570445999, −7.299000227855195, −6.498221514546552, −5.752340917974148, −5.189491629909670, −4.414148679686017, −3.690348301258547, −2.984895538077002, −2.546720698573449, −1.844728002628330, −1.099188308814597, 0, 1.099188308814597, 1.844728002628330, 2.546720698573449, 2.984895538077002, 3.690348301258547, 4.414148679686017, 5.189491629909670, 5.752340917974148, 6.498221514546552, 7.299000227855195, 7.658481570445999, 7.998637566918170, 8.744288985594174, 9.037420426536510, 9.616989158560158, 10.24437427913259, 10.48565238189661, 11.33422240767362, 11.84585823184490, 12.50760655863588, 12.66759575988296, 13.69057364271465, 14.18120843858814, 14.35415004838895, 14.92895295022713

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