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  :  $\chi_{735} (607, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ 0.615 + 0.787i)\)
\(L(1)\)  \(\approx\)  \(4.14833 - 2.02309i\)
\(L(\frac12)\)  \(\approx\)  \(4.14833 - 2.02309i\)
\(L(\frac{3}{2})\)   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

−10.52968583245611786390575110352, −9.912286899976859310799765638771, −8.332848925289633877322425158298, −7.03374407316003476326806302607, −6.60543636290283406776419964886, −5.77594348608838464747427338023, −4.69128467199065618313903641236, −3.64626456868849468321088795107, −2.65959239798027033279835629081, −1.86868576307244284055008740055, 2.08388144451267314755877085131, 3.34584004630586620895434574896, 4.37571661358874332018231606352, 4.88376725684185815033255053955, 5.84082135064352923032561509343, 6.57090841926762649873117542257, 7.74617431291867757098837824997, 8.667637762534506061912250288229, 9.606978558095051005870107101787, 10.82566287167097479122959789849

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