Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $-0.795 + 0.605i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.737 − 0.197i)2-s + (−0.258 − 0.965i)3-s + (−1.22 − 0.708i)4-s + (1.19 − 1.88i)5-s + 0.763i·6-s + (1.84 + 1.84i)8-s + (−0.866 + 0.499i)9-s + (−1.25 + 1.15i)10-s + (1.92 − 3.33i)11-s + (−0.366 + 1.36i)12-s + (3.66 − 3.66i)13-s + (−2.13 − 0.668i)15-s + (0.419 + 0.726i)16-s + (2.03 − 0.545i)17-s + (0.737 − 0.197i)18-s + (0.0348 + 0.0604i)19-s + ⋯
L(s)  = 1  + (−0.521 − 0.139i)2-s + (−0.149 − 0.557i)3-s + (−0.613 − 0.354i)4-s + (0.535 − 0.844i)5-s + 0.311i·6-s + (0.652 + 0.652i)8-s + (−0.288 + 0.166i)9-s + (−0.397 + 0.365i)10-s + (0.580 − 1.00i)11-s + (−0.105 + 0.394i)12-s + (1.01 − 1.01i)13-s + (−0.550 − 0.172i)15-s + (0.104 + 0.181i)16-s + (0.493 − 0.132i)17-s + (0.173 − 0.0465i)18-s + (0.00800 + 0.0138i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.795 + 0.605i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (178, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ -0.795 + 0.605i)\)
\(L(1)\)  \(\approx\)  \(0.319063 - 0.946254i\)
\(L(\frac12)\)  \(\approx\)  \(0.319063 - 0.946254i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-1.19 + 1.88i)T \)
7 \( 1 \)
good2 \( 1 + (0.737 + 0.197i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-1.92 + 3.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.66 + 3.66i)T - 13iT^{2} \)
17 \( 1 + (-2.03 + 0.545i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.0348 - 0.0604i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.195 - 0.729i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.77iT - 29T^{2} \)
31 \( 1 + (-2.07 - 1.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.45 + 2.26i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.68iT - 41T^{2} \)
43 \( 1 + (2.77 + 2.77i)T + 43iT^{2} \)
47 \( 1 + (2.00 - 7.50i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (8.38 - 2.24i)T + (45.8 - 26.5i)T^{2} \)
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 \( 1 + (-0.152 - 0.568i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.12T + 71T^{2} \)
73 \( 1 + (3.49 + 13.0i)T + (-63.2 + 36.5i)T^{2} \)
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

−9.962952816053530686516338948645, −9.062795522015082955696556448640, −8.447583879531866553786529010813, −7.83436291296505141169874525783, −6.23667394238161662887143609834, −5.70165680025970647451637068741, −4.75351557926279581168183418843, −3.35934159780779009611583261589, −1.56819394968077822129608119758, −0.71072741413049919656289534233, 1.70744502070443073746881345771, 3.42448947775880675093793782741, 4.18602362026133518947018311525, 5.31093441471487996090401573667, 6.57539392132479639659119021864, 7.12687325064121278553224825895, 8.384007644915022839910386785444, 9.120531246006126201753139040770, 9.851699089895906966921176711439, 10.39332775567184799421963894819

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