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

Related objects

Downloads

Learn more about

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  =  \((2,\ 735,\ (\ :1/2),\ 0.615 - 0.787i)\)
\(L(1)\)  \(\approx\)  \(4.14833 + 2.02309i\)
\(L(\frac12)\)  \(\approx\)  \(4.14833 + 2.02309i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-8.68 - 8.68i)T + 97iT^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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