Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.236·2-s − 3·3-s − 7.94·4-s + 5·5-s − 0.708·6-s − 3.76·8-s + 9·9-s + 1.18·10-s − 50.4·11-s + 23.8·12-s + 80.9·13-s − 15·15-s + 62.6·16-s − 76.3·17-s + 2.12·18-s − 4.13·19-s − 39.7·20-s − 11.9·22-s − 204.·23-s + 11.2·24-s + 25·25-s + 19.1·26-s − 27·27-s − 91.1·29-s − 3.54·30-s − 198.·31-s + 44.9·32-s + ⋯
L(s)  = 1  + 0.0834·2-s − 0.577·3-s − 0.993·4-s + 0.447·5-s − 0.0481·6-s − 0.166·8-s + 0.333·9-s + 0.0373·10-s − 1.38·11-s + 0.573·12-s + 1.72·13-s − 0.258·15-s + 0.979·16-s − 1.08·17-s + 0.0278·18-s − 0.0499·19-s − 0.444·20-s − 0.115·22-s − 1.85·23-s + 0.0960·24-s + 0.200·25-s + 0.144·26-s − 0.192·27-s − 0.583·29-s − 0.0215·30-s − 1.14·31-s + 0.248·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{735} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.014644887\)
\(L(\frac12)\)  \(\approx\)  \(1.014644887\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + 3T \)
5 \( 1 - 5T \)
7 \( 1 \)
good2 \( 1 - 0.236T + 8T^{2} \)
11 \( 1 + 50.4T + 1.33e3T^{2} \)
13 \( 1 - 80.9T + 2.19e3T^{2} \)
17 \( 1 + 76.3T + 4.91e3T^{2} \)
19 \( 1 + 4.13T + 6.85e3T^{2} \)
23 \( 1 + 204.T + 1.21e4T^{2} \)
29 \( 1 + 91.1T + 2.43e4T^{2} \)
31 \( 1 + 198.T + 2.97e4T^{2} \)
37 \( 1 - 155.T + 5.06e4T^{2} \)
41 \( 1 - 156.T + 6.89e4T^{2} \)
43 \( 1 - 354.T + 7.95e4T^{2} \)
47 \( 1 - 175.T + 1.03e5T^{2} \)
53 \( 1 - 200.T + 1.48e5T^{2} \)
59 \( 1 - 312.T + 2.05e5T^{2} \)
61 \( 1 - 154.T + 2.26e5T^{2} \)
67 \( 1 - 734.T + 3.00e5T^{2} \)
71 \( 1 + 678.T + 3.57e5T^{2} \)
73 \( 1 - 60.8T + 3.89e5T^{2} \)
79 \( 1 + 1.28e3T + 4.93e5T^{2} \)
83 \( 1 + 116.T + 5.71e5T^{2} \)
89 \( 1 - 916.T + 7.04e5T^{2} \)
97 \( 1 - 1.41e3T + 9.12e5T^{2} \)
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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

−10.09822262855160232908277404681, −9.101716728312629157979502679994, −8.393794864784479190534610305826, −7.46432206005037281795695406249, −5.98002292838930365049363290212, −5.72722030850673878027078542275, −4.52672211296869160804231260173, −3.70316437819187515230033112303, −2.10565151223534747762414395514, −0.57884463335329817099894324883, 0.57884463335329817099894324883, 2.10565151223534747762414395514, 3.70316437819187515230033112303, 4.52672211296869160804231260173, 5.72722030850673878027078542275, 5.98002292838930365049363290212, 7.46432206005037281795695406249, 8.393794864784479190534610305826, 9.101716728312629157979502679994, 10.09822262855160232908277404681

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