Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.492 + 0.870i$
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 + (4.80 − 1.39i)5-s + 2.44·6-s + (1.87 − 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s + (−3.40 + 6.19i)10-s − 11.5·11-s + (−2.44 + 2.44i)12-s + (−7.70 − 7.70i)13-s + 3.74i·14-s + (−7.58 − 4.17i)15-s − 4·16-s + (21.0 − 21.0i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (0.960 − 0.278i)5-s + 0.408·6-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.340 + 0.619i)10-s − 1.04·11-s + (−0.204 + 0.204i)12-s + (−0.592 − 0.592i)13-s + 0.267i·14-s + (−0.505 − 0.278i)15-s − 0.250·16-s + (1.23 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.492 + 0.870i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.492 + 0.870i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.931334 - 0.543335i\)
\(L(\frac12)\)  \(\approx\)  \(0.931334 - 0.543335i\)
\(L(2)\)   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 \( 1 + (-4.80 + 1.39i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good11 \( 1 + 11.5T + 121T^{2} \)
13 \( 1 + (7.70 + 7.70i)T + 169iT^{2} \)
17 \( 1 + (-21.0 + 21.0i)T - 289iT^{2} \)
19 \( 1 + 24.1iT - 361T^{2} \)
23 \( 1 + (-30.1 - 30.1i)T + 529iT^{2} \)
29 \( 1 + 51.1iT - 841T^{2} \)
31 \( 1 + 46.9T + 961T^{2} \)
37 \( 1 + (8.50 - 8.50i)T - 1.36e3iT^{2} \)
41 \( 1 - 18.6T + 1.68e3T^{2} \)
43 \( 1 + (26.1 + 26.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-50.1 + 50.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (-7.08 - 7.08i)T + 2.80e3iT^{2} \)
59 \( 1 - 94.3iT - 3.48e3T^{2} \)
61 \( 1 - 8.09T + 3.72e3T^{2} \)
67 \( 1 + (20.6 - 20.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 63.7T + 5.04e3T^{2} \)
73 \( 1 + (-50.1 - 50.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 1.06iT - 6.24e3T^{2} \)
83 \( 1 + (-53.6 - 53.6i)T + 6.88e3iT^{2} \)
89 \( 1 - 145. iT - 7.92e3T^{2} \)
97 \( 1 + (-23.7 + 23.7i)T - 9.40e3iT^{2} \)
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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

−11.93170513306617112267701835279, −10.82420278810676377136755001904, −9.932570222275736754102023195828, −9.086951084212258547638034249087, −7.68243331441464452872698828847, −7.10409530697256218830369024145, −5.49673435462440286461742666346, −5.17682170972176716509056829521, −2.55350167610511474621094216171, −0.77416417535641666486886929283, 1.73302924180320869885019274010, 3.22072595984949487759906627580, 4.96301694933053890835543990686, 5.94873926464932817921821782818, 7.32175817058413922982230268091, 8.581716211478369072078735691702, 9.579737877207387121272292173076, 10.50988589044665361683695306298, 10.87861518090276755092428288860, 12.45433372189491791655985774374

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