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 − 5·11-s + 12-s + 5·13-s + 16-s + 4·17-s − 18-s − 7·19-s + 5·22-s − 23-s − 24-s − 5·26-s + 27-s − 2·31-s − 32-s − 5·33-s − 4·34-s + 36-s − 37-s + 7·38-s + 5·39-s + 5·41-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 − 1.50·11-s + 0.288·12-s + 1.38·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.60·19-s + 1.06·22-s − 0.208·23-s − 0.204·24-s − 0.980·26-s + 0.192·27-s − 0.359·31-s − 0.176·32-s − 0.870·33-s − 0.685·34-s + 1/6·36-s − 0.164·37-s + 1.13·38-s + 0.800·39-s + 0.780·41-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  =  \((2,\ 7350,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.584718749\)
\(L(\frac12)\)  \(\approx\)  \(1.584718749\)
\(L(\frac{3}{2})\)   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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−8.061335777140358874427989036431, −7.45209403450699732441734234629, −6.64866253619909322705935432199, −5.88688890201911569867912998147, −5.24110735104902596883572679458, −4.12632377956553153558704731098, −3.43274398354345867434228407653, −2.54652796326024378385988856003, −1.84446670648064812681143920533, −0.67699326952866600560058873068, 0.67699326952866600560058873068, 1.84446670648064812681143920533, 2.54652796326024378385988856003, 3.43274398354345867434228407653, 4.12632377956553153558704731098, 5.24110735104902596883572679458, 5.88688890201911569867912998147, 6.64866253619909322705935432199, 7.45209403450699732441734234629, 8.061335777140358874427989036431

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