Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.926 + 0.376i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.867 − 0.867i)2-s + (1.22 − 1.22i)3-s − 2.49i·4-s + (−4.93 − 0.833i)5-s − 2.12·6-s + (−1.87 − 1.87i)7-s + (−5.63 + 5.63i)8-s − 2.99i·9-s + (3.55 + 5.00i)10-s − 1.49·11-s + (−3.05 − 3.05i)12-s + (2.15 − 2.15i)13-s + 3.24i·14-s + (−7.05 + 5.01i)15-s − 0.198·16-s + (−2.96 − 2.96i)17-s + ⋯
L(s)  = 1  + (−0.433 − 0.433i)2-s + (0.408 − 0.408i)3-s − 0.623i·4-s + (−0.986 − 0.166i)5-s − 0.354·6-s + (−0.267 − 0.267i)7-s + (−0.704 + 0.704i)8-s − 0.333i·9-s + (0.355 + 0.500i)10-s − 0.136·11-s + (−0.254 − 0.254i)12-s + (0.165 − 0.165i)13-s + 0.231i·14-s + (−0.470 + 0.334i)15-s − 0.0124·16-s + (−0.174 − 0.174i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.376i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.926 + 0.376i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.926 + 0.376i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.150467 - 0.769723i\)
\(L(\frac12)\)  \(\approx\)  \(0.150467 - 0.769723i\)
\(L(2)\)   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 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (4.93 + 0.833i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good2 \( 1 + (0.867 + 0.867i)T + 4iT^{2} \)
11 \( 1 + 1.49T + 121T^{2} \)
13 \( 1 + (-2.15 + 2.15i)T - 169iT^{2} \)
17 \( 1 + (2.96 + 2.96i)T + 289iT^{2} \)
19 \( 1 + 34.8iT - 361T^{2} \)
23 \( 1 + (7.50 - 7.50i)T - 529iT^{2} \)
29 \( 1 + 37.1iT - 841T^{2} \)
31 \( 1 - 47.0T + 961T^{2} \)
37 \( 1 + (-16.3 - 16.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 73.4T + 1.68e3T^{2} \)
43 \( 1 + (0.244 - 0.244i)T - 1.84e3iT^{2} \)
47 \( 1 + (38.9 + 38.9i)T + 2.20e3iT^{2} \)
53 \( 1 + (33.0 - 33.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 31.6iT - 3.48e3T^{2} \)
61 \( 1 + 106.T + 3.72e3T^{2} \)
67 \( 1 + (-28.6 - 28.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 15.8T + 5.04e3T^{2} \)
73 \( 1 + (26.2 - 26.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 73.8iT - 6.24e3T^{2} \)
83 \( 1 + (-58.6 + 58.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 83.2iT - 7.92e3T^{2} \)
97 \( 1 + (103. + 103. i)T + 9.40e3iT^{2} \)
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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

−13.06585832411293152576676543266, −11.77227089892188865434537333361, −11.06147108458043329113279329090, −9.778855803807633689598212151881, −8.773876372169554628037039904125, −7.67747546975563598404553050535, −6.36677280116450185507157891035, −4.59419667364649356100759367736, −2.76831756049325157203915062548, −0.64703589347616853402397174321, 3.13346738144325265018058874744, 4.24584778593444735132790089421, 6.31825271511551984716725501830, 7.69843327964140181229163852842, 8.331229543296441072907610466464, 9.431135251902098506280262582438, 10.72028037825664761776586222910, 12.03215527600930041373904298650, 12.72823402428422311575839938648, 14.24413725657792326577815073690

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