Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.420 + 0.907i$
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 + (−2.14 − 2.09i)3-s + 2i·4-s + (−1.13 + 4.86i)5-s + (0.0523 + 4.24i)6-s + (6.60 − 2.31i)7-s + (2 − 2i)8-s + (0.222 + 8.99i)9-s + (6.00 − 3.73i)10-s − 16.9i·11-s + (4.18 − 4.29i)12-s + (10.2 − 10.2i)13-s + (−8.91 − 4.29i)14-s + (12.6 − 8.07i)15-s − 4·16-s + (−8.79 + 8.79i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.715 − 0.698i)3-s + 0.5i·4-s + (−0.227 + 0.973i)5-s + (0.00872 + 0.707i)6-s + (0.943 − 0.330i)7-s + (0.250 − 0.250i)8-s + (0.0246 + 0.999i)9-s + (0.600 − 0.373i)10-s − 1.54i·11-s + (0.349 − 0.357i)12-s + (0.786 − 0.786i)13-s + (−0.637 − 0.306i)14-s + (0.842 − 0.538i)15-s − 0.250·16-s + (−0.517 + 0.517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.420 + 0.907i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.420 + 0.907i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.448523 - 0.702095i\)
\(L(\frac12)\)  \(\approx\)  \(0.448523 - 0.702095i\)
\(L(2)\)   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 \( 1 + (2.14 + 2.09i)T \)
5 \( 1 + (1.13 - 4.86i)T \)
7 \( 1 + (-6.60 + 2.31i)T \)
good11 \( 1 + 16.9iT - 121T^{2} \)
13 \( 1 + (-10.2 + 10.2i)T - 169iT^{2} \)
17 \( 1 + (8.79 - 8.79i)T - 289iT^{2} \)
19 \( 1 + 24.7T + 361T^{2} \)
23 \( 1 + (-19.2 + 19.2i)T - 529iT^{2} \)
29 \( 1 - 1.67T + 841T^{2} \)
31 \( 1 + 36.8iT - 961T^{2} \)
37 \( 1 + (-40.5 + 40.5i)T - 1.36e3iT^{2} \)
41 \( 1 + 0.885T + 1.68e3T^{2} \)
43 \( 1 + (9.87 + 9.87i)T + 1.84e3iT^{2} \)
47 \( 1 + (-33.7 + 33.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (11.9 - 11.9i)T - 2.80e3iT^{2} \)
59 \( 1 + 50.5iT - 3.48e3T^{2} \)
61 \( 1 + 80.6iT - 3.72e3T^{2} \)
67 \( 1 + (4.46 - 4.46i)T - 4.48e3iT^{2} \)
71 \( 1 - 137. iT - 5.04e3T^{2} \)
73 \( 1 + (53.3 - 53.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 127. iT - 6.24e3T^{2} \)
83 \( 1 + (-60.0 - 60.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 51.3iT - 7.92e3T^{2} \)
97 \( 1 + (-0.274 - 0.274i)T + 9.40e3iT^{2} \)
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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

−11.20395152325374278271190617045, −11.15058401069792534803933824156, −10.46150896630405108203215720591, −8.507921128837744161925850732811, −7.981553548270598653149966542600, −6.73174224228613617328171711121, −5.78709851419742338083275290645, −4.00480158490457079671089894079, −2.39722120399335319313365568819, −0.65251831121203243804378058541, 1.51266329987602417101582161219, 4.45838350726651545026427368777, 4.85503380194748373993970176303, 6.21432347021428355222593102154, 7.42904899425080802913388498616, 8.772323632487183643251610337581, 9.245104815096424771992821242640, 10.46457141135271506804877977747, 11.44904671549263440043837459728, 12.16256768831074150513367873406

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