Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \cdot 17 $
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 − 5-s + 6-s + 8-s + 9-s − 10-s + 12-s + 13-s − 15-s + 16-s + 17-s + 18-s − 2·19-s − 20-s − 8·23-s + 24-s + 25-s + 26-s + 27-s + 6·29-s − 30-s + 4·31-s + 32-s + 34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.458·19-s − 0.223·20-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 1.11·29-s − 0.182·30-s + 0.718·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324870\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{324870} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324870,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(5.349347794\)
\(L(\frac12)\)  \(\approx\)  \(5.349347794\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 - T \)
17 \( 1 - T \)
good11 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−12.62074480736846, −12.15854066789823, −11.77143558528603, −11.51479375588507, −10.70915534925815, −10.41418747438169, −9.937010174104211, −9.517775632981924, −8.793689993943226, −8.323068751612567, −8.034011338696631, −7.660094796652467, −6.834644728277116, −6.621822470094058, −6.117786087834943, −5.492051949202199, −4.941289372915363, −4.460481471435286, −3.937175450047186, −3.558418626566505, −3.063418955920156, −2.274895541811257, −2.081969032848174, −1.172035608490579, −0.5304284600217596, 0.5304284600217596, 1.172035608490579, 2.081969032848174, 2.274895541811257, 3.063418955920156, 3.558418626566505, 3.937175450047186, 4.460481471435286, 4.941289372915363, 5.492051949202199, 6.117786087834943, 6.621822470094058, 6.834644728277116, 7.660094796652467, 8.034011338696631, 8.323068751612567, 8.793689993943226, 9.517775632981924, 9.937010174104211, 10.41418747438169, 10.70915534925815, 11.51479375588507, 11.77143558528603, 12.15854066789823, 12.62074480736846

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