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 − 12-s − 5·13-s + 16-s + 6·17-s + 18-s − 7·19-s + 6·23-s − 24-s − 5·26-s − 27-s + 8·31-s + 32-s + 6·34-s + 36-s + 37-s − 7·38-s + 5·39-s − 8·43-s + 6·46-s − 6·47-s − 48-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 − 1.38·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.60·19-s + 1.25·23-s − 0.204·24-s − 0.980·26-s − 0.192·27-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.164·37-s − 1.13·38-s + 0.800·39-s − 1.21·43-s + 0.884·46-s − 0.875·47-s − 0.144·48-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  =  \((2,\ 7350,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.410562430\)
\(L(\frac12)\)  \(\approx\)  \(2.410562430\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−7.78602661189941316189381600394, −6.96701023253412025401166352835, −6.51379397416272804237803037266, −5.72025581776550106020594538232, −4.93585099272559953846157216901, −4.64138751314449031630956991635, −3.59502208153966225795409474985, −2.80023663048521995966224851718, −1.92111636972982141682275784635, −0.71656322584999134424247059485, 0.71656322584999134424247059485, 1.92111636972982141682275784635, 2.80023663048521995966224851718, 3.59502208153966225795409474985, 4.64138751314449031630956991635, 4.93585099272559953846157216901, 5.72025581776550106020594538232, 6.51379397416272804237803037266, 6.96701023253412025401166352835, 7.78602661189941316189381600394

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