Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.772 + 0.634i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (−2.20 + 4.48i)5-s − 2.44·6-s + (−1.87 − 1.87i)7-s + (2 − 2i)8-s − 2.99i·9-s + (6.69 − 2.28i)10-s + 15.1·11-s + (2.44 + 2.44i)12-s + (14.6 − 14.6i)13-s + 3.74i·14-s + (2.79 + 8.19i)15-s − 4·16-s + (15.9 + 15.9i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (−0.440 + 0.897i)5-s − 0.408·6-s + (−0.267 − 0.267i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.669 − 0.228i)10-s + 1.37·11-s + (0.204 + 0.204i)12-s + (1.12 − 1.12i)13-s + 0.267i·14-s + (0.186 + 0.546i)15-s − 0.250·16-s + (0.940 + 0.940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.772 + 0.634i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.772 + 0.634i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.29438 - 0.463700i\)
\(L(\frac12)\)  \(\approx\)  \(1.29438 - 0.463700i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (2.20 - 4.48i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good11 \( 1 - 15.1T + 121T^{2} \)
13 \( 1 + (-14.6 + 14.6i)T - 169iT^{2} \)
17 \( 1 + (-15.9 - 15.9i)T + 289iT^{2} \)
19 \( 1 - 2.05iT - 361T^{2} \)
23 \( 1 + (-19.4 + 19.4i)T - 529iT^{2} \)
29 \( 1 - 33.1iT - 841T^{2} \)
31 \( 1 - 27.8T + 961T^{2} \)
37 \( 1 + (30.4 + 30.4i)T + 1.36e3iT^{2} \)
41 \( 1 + 26.6T + 1.68e3T^{2} \)
43 \( 1 + (-7.89 + 7.89i)T - 1.84e3iT^{2} \)
47 \( 1 + (-33.8 - 33.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (-5.60 + 5.60i)T - 2.80e3iT^{2} \)
59 \( 1 - 5.80iT - 3.48e3T^{2} \)
61 \( 1 + 98.2T + 3.72e3T^{2} \)
67 \( 1 + (-51.6 - 51.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 120.T + 5.04e3T^{2} \)
73 \( 1 + (-81.9 + 81.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 33.0iT - 6.24e3T^{2} \)
83 \( 1 + (97.0 - 97.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 34.2iT - 7.92e3T^{2} \)
97 \( 1 + (-104. - 104. i)T + 9.40e3iT^{2} \)
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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

−12.04009822949052183486755338867, −10.87006115413058602387898382764, −10.31611116667279847641342614471, −8.972827996937529554496023282348, −8.140303882436260153400464077517, −7.07767013631261688548329478784, −6.15130875527573142579735626182, −3.83884350791829336078430614737, −3.09618263054485291305880923166, −1.18782990679096045559197901198, 1.29110722981586088033091311054, 3.61164725923202393317325127270, 4.76913078011863874158884449176, 6.12662398681412039814943057425, 7.30522758611349023075280055950, 8.561037829980350731979422758431, 9.106197214439302199704191040636, 9.839342055412793127791636326998, 11.45033030324746259598770959924, 11.95276997463105068185926040555

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