Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7^{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 − 4·11-s − 12-s − 13-s + 16-s − 2·17-s + 18-s − 19-s − 4·22-s − 2·23-s − 24-s − 26-s − 27-s + 4·29-s + 32-s + 4·33-s − 2·34-s + 36-s + 3·37-s − 38-s + 39-s − 12·41-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 − 1.20·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.229·19-s − 0.852·22-s − 0.417·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.742·29-s + 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s + 0.493·37-s − 0.162·38-s + 0.160·39-s − 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7350} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7350,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.041979134\)
\(L(\frac12)\)  \(\approx\)  \(2.041979134\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−7.893909511025408414009744268015, −6.93880657598779989753774222994, −6.53238272261620612800279204509, −5.59547500730985464441344334863, −5.17656950414959687428838247745, −4.45674095074955608210302249626, −3.69572210888079422895223509190, −2.67227497408484362684533794835, −2.03785763818337133645936034673, −0.64146197935357057367811881592, 0.64146197935357057367811881592, 2.03785763818337133645936034673, 2.67227497408484362684533794835, 3.69572210888079422895223509190, 4.45674095074955608210302249626, 5.17656950414959687428838247745, 5.59547500730985464441344334863, 6.53238272261620612800279204509, 6.93880657598779989753774222994, 7.893909511025408414009744268015

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