Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $-0.511 + 0.859i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.681 − 2.54i)2-s + (−0.965 − 0.258i)3-s + (−4.26 + 2.46i)4-s + (−2.18 + 0.478i)5-s + 2.63i·6-s + (5.45 + 5.45i)8-s + (0.866 + 0.499i)9-s + (2.70 + 5.22i)10-s + (0.731 + 1.26i)11-s + (4.76 − 1.27i)12-s + (−0.887 + 0.887i)13-s + (2.23 + 0.103i)15-s + (5.22 − 9.04i)16-s + (0.770 − 2.87i)17-s + (0.681 − 2.54i)18-s + (−1.97 + 3.42i)19-s + ⋯
L(s)  = 1  + (−0.481 − 1.79i)2-s + (−0.557 − 0.149i)3-s + (−2.13 + 1.23i)4-s + (−0.976 + 0.213i)5-s + 1.07i·6-s + (1.92 + 1.92i)8-s + (0.288 + 0.166i)9-s + (0.855 + 1.65i)10-s + (0.220 + 0.381i)11-s + (1.37 − 0.368i)12-s + (−0.246 + 0.246i)13-s + (0.576 + 0.0267i)15-s + (1.30 − 2.26i)16-s + (0.186 − 0.697i)17-s + (0.160 − 0.599i)18-s + (−0.454 + 0.786i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.511 + 0.859i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (313, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ -0.511 + 0.859i)\)
\(L(1)\)  \(\approx\)  \(0.292030 - 0.513458i\)
\(L(\frac12)\)  \(\approx\)  \(0.292030 - 0.513458i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (2.18 - 0.478i)T \)
7 \( 1 \)
good2 \( 1 + (0.681 + 2.54i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-0.731 - 1.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.887 - 0.887i)T - 13iT^{2} \)
17 \( 1 + (-0.770 + 2.87i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.97 - 3.42i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.64 + 1.51i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.18iT - 29T^{2} \)
31 \( 1 + (5.28 - 3.05i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.825 - 3.08i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.769iT - 41T^{2} \)
43 \( 1 + (5.18 + 5.18i)T + 43iT^{2} \)
47 \( 1 + (-11.7 + 3.13i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.199 - 0.743i)T + (-45.8 - 26.5i)T^{2} \)
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 \( 1 + (-8.10 - 2.17i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.62T + 71T^{2} \)
73 \( 1 + (-9.30 - 2.49i)T + (63.2 + 36.5i)T^{2} \)
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

−10.37941078519757666788976130074, −9.504346798101598635434332531567, −8.651867909649730382028980685012, −7.74006931630986253397400599732, −6.82402077443530510525081212299, −5.11845077036916619035278626238, −4.23400561591974228275620170968, −3.36128578620864624421919994052, −2.13649980977366905223447193527, −0.67675534729583733062235122390, 0.74721852781424225259354661273, 3.69017817028017394209479924662, 4.73955295323397279001741569169, 5.43705033121786655291731905497, 6.44752806455078975984765530094, 7.20218769007066744735598848090, 7.892883998450100542455724279836, 8.812990711923029458806890270724, 9.337223339653441817688349008564, 10.59948622664722261547061799914

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