Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.0402 + 0.999i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (−1.93 + 1.11i)5-s + 2.44i·6-s + (5.73 + 4.01i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (2.73 + 1.58i)10-s + (8.69 − 15.0i)11-s + (2.99 − 1.73i)12-s − 7.22i·13-s + (0.854 − 9.86i)14-s + 3.87·15-s + (−2.00 − 3.46i)16-s + (−2.29 − 1.32i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + 0.408i·6-s + (0.819 + 0.572i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.273 + 0.158i)10-s + (0.790 − 1.36i)11-s + (0.249 − 0.144i)12-s − 0.555i·13-s + (0.0610 − 0.704i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (−0.135 − 0.0779i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0402 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0402 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.0402 + 0.999i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.0402 + 0.999i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.752885 - 0.723141i\)
\(L(\frac12)\)  \(\approx\)  \(0.752885 - 0.723141i\)
\(L(2)\)   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 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (-5.73 - 4.01i)T \)
good11 \( 1 + (-8.69 + 15.0i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 7.22iT - 169T^{2} \)
17 \( 1 + (2.29 + 1.32i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.20 + 1.27i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (20.0 + 34.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 47.0T + 841T^{2} \)
31 \( 1 + (-34.9 - 20.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (16.2 + 28.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 70.6iT - 1.68e3T^{2} \)
43 \( 1 - 37.3T + 1.84e3T^{2} \)
47 \( 1 + (28.9 - 16.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-35.4 + 61.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-87.0 - 50.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (11.1 - 6.44i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (47.0 - 81.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 11.5T + 5.04e3T^{2} \)
73 \( 1 + (19.6 + 11.3i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-12.0 - 20.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 111. iT - 6.88e3T^{2} \)
89 \( 1 + (110. - 63.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 7.48iT - 9.40e3T^{2} \)
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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

−11.89170073649788804270807482133, −11.01636013493599001233785381452, −10.29333155472151383123408595806, −8.659833409733004941781162265741, −8.268711427346646391535152367254, −6.78762899743981239999310920423, −5.59832019811542215953411403572, −4.19519222153596210898625415732, −2.63672782757444040459692455821, −0.824193187289460405166805549776, 1.40788048862622439143579048612, 4.18480284090674702097134680129, 4.83343761933332817433607051703, 6.34939473564642098832914702574, 7.32132872059255493378855534610, 8.244366942216345544809137377110, 9.520079442167412456698504644501, 10.22344836556073530999180742753, 11.54130486365903420401885792189, 12.02810724735135133782372093212

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