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  :  $\chi_{735} (313, \cdot )$
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

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

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