Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.0673 - 0.997i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0859 + 0.148i)2-s + (0.346 + 2.97i)3-s + (1.98 + 3.43i)4-s + (4.85 − 1.19i)5-s + (−0.473 − 0.204i)6-s + (−6.99 + 0.246i)7-s − 1.37·8-s + (−8.75 + 2.06i)9-s + (−0.239 + 0.825i)10-s + (10.0 − 5.79i)11-s + (−9.55 + 7.10i)12-s + 7.34i·13-s + (0.564 − 1.06i)14-s + (5.25 + 14.0i)15-s + (−7.82 + 13.5i)16-s + (2.30 + 3.99i)17-s + ⋯
L(s)  = 1  + (−0.0429 + 0.0744i)2-s + (0.115 + 0.993i)3-s + (0.496 + 0.859i)4-s + (0.970 − 0.239i)5-s + (−0.0789 − 0.0340i)6-s + (−0.999 + 0.0351i)7-s − 0.171·8-s + (−0.973 + 0.229i)9-s + (−0.0239 + 0.0825i)10-s + (0.913 − 0.527i)11-s + (−0.796 + 0.592i)12-s + 0.565i·13-s + (0.0403 − 0.0759i)14-s + (0.350 + 0.936i)15-s + (−0.488 + 0.846i)16-s + (0.135 + 0.234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0673 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0673 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.0673 - 0.997i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (44, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.0673 - 0.997i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.13990 + 1.06553i\)
\(L(\frac12)\)  \(\approx\)  \(1.13990 + 1.06553i\)
\(L(2)\)   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 \( 1 + (-0.346 - 2.97i)T \)
5 \( 1 + (-4.85 + 1.19i)T \)
7 \( 1 + (6.99 - 0.246i)T \)
good2 \( 1 + (0.0859 - 0.148i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (-10.0 + 5.79i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 7.34iT - 169T^{2} \)
17 \( 1 + (-2.30 - 3.99i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.93 + 10.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-11.8 + 20.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 32.7iT - 841T^{2} \)
31 \( 1 + (23.4 + 40.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-41.2 - 23.8i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 70.7iT - 1.68e3T^{2} \)
43 \( 1 - 14.1iT - 1.84e3T^{2} \)
47 \( 1 + (-26.1 + 45.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-11.5 - 19.9i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (1.09 - 0.633i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (32.3 - 55.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-17.1 + 9.91i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 48.1iT - 5.04e3T^{2} \)
73 \( 1 + (107. - 62.2i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (34.4 - 59.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 35.8T + 6.88e3T^{2} \)
89 \( 1 + (110. + 63.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 26.9iT - 9.40e3T^{2} \)
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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

−13.76084473075098475631909991479, −12.77746564858419784267018764365, −11.61330281271630084842947914635, −10.51276171069933977608086428145, −9.278376518414389226989257613930, −8.773728666224266492351064879030, −6.87982665043315125581054342916, −5.82326082133561972025476632868, −4.04104736467289924633037959999, −2.74677787903315713023406485144, 1.38184914916396616131950132354, 2.88802039458357135207938198199, 5.64573389093354605522294223367, 6.41119030888307943071451391855, 7.33148137002081799339949171210, 9.251380192955219964217133935939, 9.910257776348089829866066953646, 11.19514852604852227908906693321, 12.35754407709575979487046585098, 13.31108102871665630785512016038

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