Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.260 - 0.965i$
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 + (−4.39 + 2.37i)5-s − 2.44·6-s + (1.87 + 1.87i)7-s + (2 − 2i)8-s − 2.99i·9-s + (6.77 + 2.01i)10-s − 14.7·11-s + (2.44 + 2.44i)12-s + (−13.2 + 13.2i)13-s − 3.74i·14-s + (−2.47 + 8.29i)15-s − 4·16-s + (1.17 + 1.17i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (−0.879 + 0.475i)5-s − 0.408·6-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.677 + 0.201i)10-s − 1.34·11-s + (0.204 + 0.204i)12-s + (−1.02 + 1.02i)13-s − 0.267i·14-s + (−0.164 + 0.553i)15-s − 0.250·16-s + (0.0693 + 0.0693i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.260 - 0.965i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.260 - 0.965i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.266607 + 0.348204i\)
\(L(\frac12)\)  \(\approx\)  \(0.266607 + 0.348204i\)
\(L(2)\)   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 \( 1 + (4.39 - 2.37i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good11 \( 1 + 14.7T + 121T^{2} \)
13 \( 1 + (13.2 - 13.2i)T - 169iT^{2} \)
17 \( 1 + (-1.17 - 1.17i)T + 289iT^{2} \)
19 \( 1 - 15.1iT - 361T^{2} \)
23 \( 1 + (22.5 - 22.5i)T - 529iT^{2} \)
29 \( 1 + 3.29iT - 841T^{2} \)
31 \( 1 - 50.0T + 961T^{2} \)
37 \( 1 + (-6.15 - 6.15i)T + 1.36e3iT^{2} \)
41 \( 1 + 35.1T + 1.68e3T^{2} \)
43 \( 1 + (58.9 - 58.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (20.1 + 20.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (-10.9 + 10.9i)T - 2.80e3iT^{2} \)
59 \( 1 + 66.6iT - 3.48e3T^{2} \)
61 \( 1 + 4.45T + 3.72e3T^{2} \)
67 \( 1 + (88.4 + 88.4i)T + 4.48e3iT^{2} \)
71 \( 1 - 69.3T + 5.04e3T^{2} \)
73 \( 1 + (-58.4 + 58.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 52.8iT - 6.24e3T^{2} \)
83 \( 1 + (53.4 - 53.4i)T - 6.88e3iT^{2} \)
89 \( 1 + 21.3iT - 7.92e3T^{2} \)
97 \( 1 + (-90.3 - 90.3i)T + 9.40e3iT^{2} \)
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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

−12.03543175056950376725810346280, −11.70869271772149590333967389550, −10.41862533095038958011979876321, −9.598811553201365688856907822622, −8.096824216876334972713320935606, −7.85743472042650864130366495116, −6.59443035776818056869852264177, −4.76124452812311165233129622337, −3.30122346113530198403730377083, −2.08163893887610663699410363479, 0.26025988267906733507163642398, 2.76121276109149525847600862461, 4.50964127650495066177029163508, 5.32480491012227810097614544330, 7.13086809440841669587225946034, 8.032225869100400891965707246727, 8.523913401299746587296658734014, 9.980555310347135774340163549389, 10.53200227532842922464257535435, 11.80099091770196848890358738521

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