Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $0.897 + 0.441i$
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 + (1.23 − 1.86i)5-s − 0.999·6-s − 3i·8-s + (0.499 + 0.866i)9-s + (−0.133 + 2.23i)10-s + (3 − 5.19i)11-s + (−0.866 + 0.499i)12-s + 2i·13-s + (2 − 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 + (0.550 − 0.834i)5-s − 0.408·6-s − 1.06i·8-s + (0.166 + 0.288i)9-s + (−0.0423 + 0.705i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $0.897 + 0.441i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (214, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ 0.897 + 0.441i)\)
\(L(1)\)  \(\approx\)  \(1.20911 - 0.281066i\)
\(L(\frac12)\)  \(\approx\)  \(1.20911 - 0.281066i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.23 + 1.86i)T \)
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

−9.885264599909206996557586616423, −9.117389360958010104859468623428, −8.780962046490047657598092891631, −8.132500004322420060046584390766, −6.86525569021167544476904604132, −6.09904079893725132173085593822, −4.66792903035204144243931585552, −3.94422141290013558236588923937, −2.58839088254207221525586711313, −0.78547004674032127317424196648, 1.60509465348059248411105966523, 2.30972295193615412235595740045, 3.79627073515957656390590813272, 5.02023216424947310413203488916, 6.28791704207243977077302512699, 6.90940117126105408411304863446, 8.060788838362176163598445220243, 8.878136502471342060663551850812, 9.737592896688344095861716074873, 10.25014352707304232121663312476

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