Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $0.185 + 0.982i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (−0.500 + 0.866i)4-s + (−2.23 + 0.133i)5-s − 0.999·6-s + 3i·8-s + (0.499 + 0.866i)9-s + (−1.86 + 1.23i)10-s + (3 − 5.19i)11-s + (0.866 − 0.499i)12-s − 2i·13-s + (1.99 + i)15-s + (0.500 + 0.866i)16-s + (3.46 + 2i)17-s + (0.866 + 0.499i)18-s + (−3 − 5.19i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (−0.250 + 0.433i)4-s + (−0.998 + 0.0599i)5-s − 0.408·6-s + 1.06i·8-s + (0.166 + 0.288i)9-s + (−0.590 + 0.389i)10-s + (0.904 − 1.56i)11-s + (0.249 − 0.144i)12-s − 0.554i·13-s + (0.516 + 0.258i)15-s + (0.125 + 0.216i)16-s + (0.840 + 0.485i)17-s + (0.204 + 0.117i)18-s + (−0.688 − 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $0.185 + 0.982i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (214, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ 0.185 + 0.982i)\)
\(L(1)\)  \(\approx\)  \(0.955769 - 0.792126i\)
\(L(\frac12)\)  \(\approx\)  \(0.955769 - 0.792126i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (2.23 - 0.133i)T \)
7 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.46 + 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.8 + 8i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-5.19 - 3i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−10.68577697072770792366376381093, −9.166463234302451591497132948218, −8.291903551141189757707854612816, −7.77405131951598163362725530767, −6.52747827430576749522381841931, −5.64709156520941416091941830062, −4.49751735650570901220318794908, −3.71132493965297492633003591314, −2.78730957972345392662946025363, −0.65350587762320102125702073930, 1.33516815787608534265648526305, 3.52039805600232131296260805486, 4.46708317145231609878473002705, 4.83771595041845853617582873269, 6.19157767177446735150333500794, 6.84319662427915933036165948530, 7.75201671397994092120938549732, 8.995037694866792524507920246491, 9.834624842932364521443680217544, 10.45010657715928759960582160546

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