Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $0.901 + 0.431i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.03 − 0.544i)2-s + (−0.258 − 0.965i)3-s + (2.10 + 1.21i)4-s + (−2.22 − 0.203i)5-s + 2.10i·6-s + (−0.640 − 0.640i)8-s + (−0.866 + 0.499i)9-s + (4.41 + 1.62i)10-s + (1.33 − 2.31i)11-s + (0.629 − 2.34i)12-s + (−1.22 + 1.22i)13-s + (0.379 + 2.20i)15-s + (−1.47 − 2.55i)16-s + (−6.48 + 1.73i)17-s + (2.03 − 0.544i)18-s + (3.00 + 5.21i)19-s + ⋯
L(s)  = 1  + (−1.43 − 0.385i)2-s + (−0.149 − 0.557i)3-s + (1.05 + 0.607i)4-s + (−0.995 − 0.0912i)5-s + 0.859i·6-s + (−0.226 − 0.226i)8-s + (−0.288 + 0.166i)9-s + (1.39 + 0.514i)10-s + (0.402 − 0.697i)11-s + (0.181 − 0.677i)12-s + (−0.340 + 0.340i)13-s + (0.0979 + 0.568i)15-s + (−0.369 − 0.639i)16-s + (−1.57 + 0.421i)17-s + (0.479 − 0.128i)18-s + (0.690 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $0.901 + 0.431i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (178, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ 0.901 + 0.431i)\)
\(L(1)\)  \(\approx\)  \(0.441045 - 0.100127i\)
\(L(\frac12)\)  \(\approx\)  \(0.441045 - 0.100127i\)
\(L(\frac{3}{2})\)   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 \( 1 + (2.22 + 0.203i)T \)
7 \( 1 \)
good2 \( 1 + (2.03 + 0.544i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-1.33 + 2.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.22 - 1.22i)T - 13iT^{2} \)
17 \( 1 + (6.48 - 1.73i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.00 - 5.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0643 - 0.239i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.304iT - 29T^{2} \)
31 \( 1 + (-6.28 - 3.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.00 - 0.269i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.05iT - 41T^{2} \)
43 \( 1 + (-0.304 - 0.304i)T + 43iT^{2} \)
47 \( 1 + (0.203 - 0.760i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.81 + 1.82i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.99 - 6.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.79 + 2.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.25 + 4.68i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (3.66 + 13.6i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-9.78 + 5.64i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.88 + 4.88i)T - 83iT^{2} \)
89 \( 1 + (-3.45 - 5.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.84 - 8.84i)T + 97iT^{2} \)
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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

−10.42939882616947354843578925233, −9.254957694045910056290156225982, −8.603949350656762588278531645535, −7.964989475777345695373141787389, −7.16938834608174997234393518886, −6.30294319269541914478475712283, −4.80863812387152059780384615355, −3.52619719703344479558297180704, −2.11503581043494242109323577783, −0.798828188844792395196980466776, 0.59728747481386606212696251923, 2.57446692974971347152763666153, 4.13109429087230066603918519738, 4.87841894484535602460229666559, 6.54862210629349402110068476521, 7.11578068586339088080196849664, 7.990947004135769817036998826007, 8.797155239716087228978163974299, 9.489225565352498081268821035023, 10.18009306558034833071515079588

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