Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.935 - 0.354i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 1.36i)2-s + (−0.448 − 1.67i)3-s + (−1.73 − i)4-s + (−3.58 − 3.48i)5-s + 2.44·6-s + (3.00 + 6.32i)7-s + (2 − 1.99i)8-s + (−2.59 + 1.50i)9-s + (6.07 − 3.61i)10-s + (−1.95 + 3.38i)11-s + (−0.896 + 3.34i)12-s + (−8.93 + 8.93i)13-s + (−9.73 + 1.78i)14-s + (−4.22 + 7.55i)15-s + (1.99 + 3.46i)16-s + (−14.3 + 3.83i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.149 − 0.557i)3-s + (−0.433 − 0.250i)4-s + (−0.716 − 0.697i)5-s + 0.408·6-s + (0.428 + 0.903i)7-s + (0.250 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.607 − 0.361i)10-s + (−0.177 + 0.308i)11-s + (−0.0747 + 0.278i)12-s + (−0.687 + 0.687i)13-s + (−0.695 + 0.127i)14-s + (−0.281 + 0.503i)15-s + (0.124 + 0.216i)16-s + (−0.841 + 0.225i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.935 - 0.354i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.935 - 0.354i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0687204 + 0.375132i\)
\(L(\frac12)\)  \(\approx\)  \(0.0687204 + 0.375132i\)
\(L(2)\)   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 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 + (0.448 + 1.67i)T \)
5 \( 1 + (3.58 + 3.48i)T \)
7 \( 1 + (-3.00 - 6.32i)T \)
good11 \( 1 + (1.95 - 3.38i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (8.93 - 8.93i)T - 169iT^{2} \)
17 \( 1 + (14.3 - 3.83i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (28.0 - 16.1i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-12.3 - 3.30i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + 26.6iT - 841T^{2} \)
31 \( 1 + (17.0 - 29.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-1.67 + 6.25i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 26.0T + 1.68e3T^{2} \)
43 \( 1 + (-21.0 + 21.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (-10.7 + 40.1i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-11.7 - 43.7i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (20.7 + 12.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-8.39 - 14.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-57.0 + 15.2i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 132.T + 5.04e3T^{2} \)
73 \( 1 + (-2.29 - 8.56i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-90.6 + 52.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-94.1 + 94.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (43.2 - 24.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-118. - 118. i)T + 9.40e3iT^{2} \)
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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

−12.47048621708434667278376235620, −11.89424353991700065931902930642, −10.70318778852351185326902255761, −9.155858865637361888742869696791, −8.515324620974151126245364802251, −7.60442548019016149114033388795, −6.52040124007613772174017927591, −5.29946941481824969180251598375, −4.28985958509058440466537631266, −1.96601886466453838492462458119, 0.22289180705527761589741784040, 2.68049474369848652003316312835, 3.96874892064080638044854840745, 4.88393164668434567971635537877, 6.72400875978410821796827575191, 7.78222903365049813278475733094, 8.826931661099843284014421837139, 10.10278697064899716404068753606, 10.97278032396449102070392610958, 11.17973912144324171517432858062

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