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 + 6·13-s + 16-s + 2·17-s − 18-s + 4·19-s + 4·22-s − 8·23-s + 24-s − 6·26-s − 27-s − 2·29-s − 32-s + 4·33-s − 2·34-s + 36-s + 10·37-s − 4·38-s − 6·39-s + 6·41-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 + 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.852·22-s − 1.66·23-s + 0.204·24-s − 1.17·26-s − 0.192·27-s − 0.371·29-s − 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s + 1.64·37-s − 0.648·38-s − 0.960·39-s + 0.937·41-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  =  \((2,\ 7350,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.066237663\)
\(L(\frac12)\)  \(\approx\)  \(1.066237663\)
\(L(\frac{3}{2})\)   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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−7.82346428638524176597481330832, −7.52281085391909472304932230967, −6.33754015492779157296733757930, −5.96351299875065035953524843895, −5.35010135518236869010586831192, −4.31089079101385664640002966612, −3.49796811497810747887778433172, −2.58069439638632572608266433034, −1.53767756142141922309330884206, −0.62442247358961096511964059650, 0.62442247358961096511964059650, 1.53767756142141922309330884206, 2.58069439638632572608266433034, 3.49796811497810747887778433172, 4.31089079101385664640002966612, 5.35010135518236869010586831192, 5.96351299875065035953524843895, 6.33754015492779157296733757930, 7.52281085391909472304932230967, 7.82346428638524176597481330832

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