Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $0.712 + 0.701i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.197 − 0.737i)2-s + (0.965 − 0.258i)3-s + (1.22 + 0.708i)4-s + (−2.23 + 0.0930i)5-s − 0.763i·6-s + (1.84 − 1.84i)8-s + (0.866 − 0.499i)9-s + (−0.373 + 1.66i)10-s + (1.92 − 3.33i)11-s + (1.36 + 0.366i)12-s + (3.66 + 3.66i)13-s + (−2.13 + 0.668i)15-s + (0.419 + 0.726i)16-s + (−0.545 − 2.03i)17-s + (−0.197 − 0.737i)18-s + (0.0348 + 0.0604i)19-s + ⋯
L(s)  = 1  + (0.139 − 0.521i)2-s + (0.557 − 0.149i)3-s + (0.613 + 0.354i)4-s + (−0.999 + 0.0416i)5-s − 0.311i·6-s + (0.652 − 0.652i)8-s + (0.288 − 0.166i)9-s + (−0.117 + 0.527i)10-s + (0.580 − 1.00i)11-s + (0.394 + 0.105i)12-s + (1.01 + 1.01i)13-s + (−0.550 + 0.172i)15-s + (0.104 + 0.181i)16-s + (−0.132 − 0.493i)17-s + (−0.0465 − 0.173i)18-s + (0.00800 + 0.0138i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $0.712 + 0.701i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (472, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ 0.712 + 0.701i)\)
\(L(1)\)  \(\approx\)  \(2.02236 - 0.827970i\)
\(L(\frac12)\)  \(\approx\)  \(2.02236 - 0.827970i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (2.23 - 0.0930i)T \)
7 \( 1 \)
good2 \( 1 + (-0.197 + 0.737i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-1.92 + 3.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.66 - 3.66i)T + 13iT^{2} \)
17 \( 1 + (0.545 + 2.03i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.0348 - 0.0604i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.729 - 0.195i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.77iT - 29T^{2} \)
31 \( 1 + (2.07 + 1.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.26 + 8.45i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.68iT - 41T^{2} \)
43 \( 1 + (2.77 - 2.77i)T - 43iT^{2} \)
47 \( 1 + (-7.50 - 2.00i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.24 - 8.38i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.48 - 6.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12.3 - 7.15i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.568 - 0.152i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.12T + 71T^{2} \)
73 \( 1 + (-13.0 + 3.49i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.54 - 4.93i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.63 + 1.63i)T + 83iT^{2} \)
89 \( 1 + (2.52 + 4.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.85 - 6.85i)T - 97iT^{2} \)
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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

−10.72597971310037292281587682238, −9.152980025791742343475327597385, −8.685311641517053669887158550914, −7.59917380286426085356647390987, −7.00510181197948118408770661437, −6.00880255669452974499601317294, −4.21596843304765240761012420760, −3.69068641641804372867643549417, −2.71548355734100927267726657410, −1.26040628810906108659559384327, 1.47889903297696870742016281359, 2.98843252607594777137155435035, 4.06883966925166292255013400639, 5.03171680246533641175807518276, 6.25813392063224865057098752879, 7.03178695722743204971644211433, 7.936189011983583874458640281381, 8.419660841745126127625044544375, 9.644480637466703659748984596093, 10.54223898458058966370789845122

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