Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.963 + 0.267i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 1.36i)2-s + (−1.22 + 1.22i)3-s − 0.258i·4-s + (3.39 + 3.66i)5-s + 3.35·6-s + (1.87 + 1.87i)7-s + (−5.82 + 5.82i)8-s − 2.99i·9-s + (0.372 − 9.66i)10-s + 17.6·11-s + (0.316 + 0.316i)12-s + (12.1 − 12.1i)13-s − 5.11i·14-s + (−8.65 − 0.333i)15-s + 14.9·16-s + (13.8 + 13.8i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.683i)2-s + (−0.408 + 0.408i)3-s − 0.0645i·4-s + (0.679 + 0.733i)5-s + 0.558·6-s + (0.267 + 0.267i)7-s + (−0.728 + 0.728i)8-s − 0.333i·9-s + (0.0372 − 0.966i)10-s + 1.60·11-s + (0.0263 + 0.0263i)12-s + (0.932 − 0.932i)13-s − 0.365i·14-s + (−0.576 − 0.0222i)15-s + 0.931·16-s + (0.816 + 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.963 + 0.267i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.963 + 0.267i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.994706 - 0.135299i\)
\(L(\frac12)\)  \(\approx\)  \(0.994706 - 0.135299i\)
\(L(2)\)   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 \( 1 + (-3.39 - 3.66i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good2 \( 1 + (1.36 + 1.36i)T + 4iT^{2} \)
11 \( 1 - 17.6T + 121T^{2} \)
13 \( 1 + (-12.1 + 12.1i)T - 169iT^{2} \)
17 \( 1 + (-13.8 - 13.8i)T + 289iT^{2} \)
19 \( 1 + 18.3iT - 361T^{2} \)
23 \( 1 + (26.3 - 26.3i)T - 529iT^{2} \)
29 \( 1 + 2.87iT - 841T^{2} \)
31 \( 1 - 16.1T + 961T^{2} \)
37 \( 1 + (-2.52 - 2.52i)T + 1.36e3iT^{2} \)
41 \( 1 + 1.89T + 1.68e3T^{2} \)
43 \( 1 + (42.5 - 42.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (57.7 + 57.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-66.5 + 66.5i)T - 2.80e3iT^{2} \)
59 \( 1 - 16.4iT - 3.48e3T^{2} \)
61 \( 1 + 7.37T + 3.72e3T^{2} \)
67 \( 1 + (27.2 + 27.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 79.5T + 5.04e3T^{2} \)
73 \( 1 + (-63.3 + 63.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 2.48iT - 6.24e3T^{2} \)
83 \( 1 + (29.0 - 29.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 29.3iT - 7.92e3T^{2} \)
97 \( 1 + (89.1 + 89.1i)T + 9.40e3iT^{2} \)
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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

−13.51912163034708873404409300255, −11.87681148412753536011259491999, −11.24211298800231376174593263310, −10.22541002161149423480669342113, −9.552335408590078008754204167638, −8.392837264262446805478947750477, −6.42688177690290701270790075894, −5.57679908987245036554020926327, −3.43697589343331234851717501521, −1.51017704379425771164674717762, 1.28539181077891087823073581093, 4.12263730492876586321356045401, 5.98657517028466195262249016021, 6.75985661350156719823778068116, 8.154333349623780298954489790491, 9.023476666628837569286933222865, 10.01827007601549604334842625605, 11.76635340843577409293534078946, 12.32091297330280131068465205139, 13.71098929129632314839177477458

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