Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $-0.997 + 0.0633i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)2-s + (1.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s − 3·6-s + 1.73i·8-s + (1.5 − 2.59i)9-s + (−1.5 + 0.866i)10-s + (−3 + 1.73i)11-s + (1.5 + 0.866i)12-s − 1.73i·15-s + (2.49 − 4.33i)16-s + (−3 − 5.19i)17-s + (−4.5 + 2.59i)18-s + (−3 − 1.73i)19-s + 20-s + ⋯
L(s)  = 1  + (−1.06 − 0.612i)2-s + (0.866 − 0.499i)3-s + (0.250 + 0.433i)4-s + (0.223 − 0.387i)5-s − 1.22·6-s + 0.612i·8-s + (0.5 − 0.866i)9-s + (−0.474 + 0.273i)10-s + (−0.904 + 0.522i)11-s + (0.433 + 0.250i)12-s − 0.447i·15-s + (0.624 − 1.08i)16-s + (−0.727 − 1.26i)17-s + (−1.06 + 0.612i)18-s + (−0.688 − 0.397i)19-s + 0.223·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.997 + 0.0633i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (656, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ -0.997 + 0.0633i)\)
\(L(1)\)  \(\approx\)  \(0.0249201 - 0.786169i\)
\(L(\frac12)\)  \(\approx\)  \(0.0249201 - 0.786169i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) 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 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (1.5 + 0.866i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 1.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−9.809771940038829417850968050782, −9.125123988989970037671485714820, −8.454232592329778857520417744689, −7.70699356168811298859583603405, −6.82708038342572479063033555453, −5.43719121558303435854774409714, −4.28606446527974744721778313483, −2.60677393596056049559788080092, −2.07819260618057647799877182396, −0.49141837750385909231436555336, 1.90854414996012034068781250975, 3.27579332436581888490403557660, 4.24135233994669874278140405937, 5.69691142222737311803911751904, 6.69400220262105208036925183015, 7.69927729188480473622791704470, 8.293573124698890403220357375671, 8.912474035453363200345648395802, 9.806687907796994709560794329683, 10.48861415732409790631254870289

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