Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $0.615 - 0.787i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.54 + 0.681i)2-s + (−0.258 − 0.965i)3-s + (4.26 + 2.46i)4-s + (−0.678 + 2.13i)5-s − 2.63i·6-s + (5.45 + 5.45i)8-s + (−0.866 + 0.499i)9-s + (−3.17 + 4.95i)10-s + (0.731 − 1.26i)11-s + (1.27 − 4.76i)12-s + (0.887 − 0.887i)13-s + (2.23 + 0.103i)15-s + (5.22 + 9.04i)16-s + (2.87 − 0.770i)17-s + (−2.54 + 0.681i)18-s + (1.97 + 3.42i)19-s + ⋯
L(s)  = 1  + (1.79 + 0.481i)2-s + (−0.149 − 0.557i)3-s + (2.13 + 1.23i)4-s + (−0.303 + 0.952i)5-s − 1.07i·6-s + (1.92 + 1.92i)8-s + (−0.288 + 0.166i)9-s + (−1.00 + 1.56i)10-s + (0.220 − 0.381i)11-s + (0.368 − 1.37i)12-s + (0.246 − 0.246i)13-s + (0.576 + 0.0267i)15-s + (1.30 + 2.26i)16-s + (0.697 − 0.186i)17-s + (−0.599 + 0.160i)18-s + (0.454 + 0.786i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $0.615 - 0.787i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (178, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ 0.615 - 0.787i)\)
\(L(1)\)  \(\approx\)  \(3.63168 + 1.77113i\)
\(L(\frac12)\)  \(\approx\)  \(3.63168 + 1.77113i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.678 - 2.13i)T \)
7 \( 1 \)
good2 \( 1 + (-2.54 - 0.681i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-0.731 + 1.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.887 + 0.887i)T - 13iT^{2} \)
17 \( 1 + (-2.87 + 0.770i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.97 - 3.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.51 - 5.64i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.18iT - 29T^{2} \)
31 \( 1 + (5.28 + 3.05i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.08 + 0.825i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.769iT - 41T^{2} \)
43 \( 1 + (5.18 + 5.18i)T + 43iT^{2} \)
47 \( 1 + (-3.13 + 11.7i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.743 + 0.199i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.59 + 2.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.23 - 0.710i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.17 + 8.10i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.62T + 71T^{2} \)
73 \( 1 + (-2.49 - 9.30i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.91 - 2.26i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.75 - 6.75i)T - 83iT^{2} \)
89 \( 1 + (0.599 + 1.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.68 + 8.68i)T + 97iT^{2} \)
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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

−11.00185158652419241587337687765, −9.931871823896958702195380084220, −8.196432251092417925267708714719, −7.47786439739476022516776053158, −6.84825019219967789220695040691, −5.87163933237028168996558055896, −5.41955172325307633082700889535, −3.83740165482266425131343545015, −3.36349369598894385220306340435, −2.08952337058427066311974092389, 1.46169110961558286673290034078, 2.98903499013679649323866656292, 3.97039049081631479710689654927, 4.71724814826682512644313031089, 5.33651705660473103178672970167, 6.30495407536270879858996995991, 7.33424500475358874730575014305, 8.662816049258066121122054214070, 9.649817882699321051724821773121, 10.63117205918323216171279198741

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