Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.0455 + 0.998i$
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 + (3.99 − 3.01i)5-s + 2.44·6-s + (−5.53 − 4.28i)7-s + (−2 + 1.99i)8-s + (−2.59 + 1.50i)9-s + (−2.65 − 6.55i)10-s + (8.47 − 14.6i)11-s + (0.896 − 3.34i)12-s + (3.05 − 3.05i)13-s + (−7.87 + 5.99i)14-s + (6.82 + 5.32i)15-s + (1.99 + 3.46i)16-s + (10.7 − 2.87i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.149 + 0.557i)3-s + (−0.433 − 0.250i)4-s + (0.798 − 0.602i)5-s + 0.408·6-s + (−0.790 − 0.612i)7-s + (−0.250 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.265 − 0.655i)10-s + (0.770 − 1.33i)11-s + (0.0747 − 0.278i)12-s + (0.234 − 0.234i)13-s + (−0.562 + 0.428i)14-s + (0.455 + 0.355i)15-s + (0.124 + 0.216i)16-s + (0.632 − 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0455 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0455 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.0455 + 0.998i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.0455 + 0.998i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.19867 - 1.25452i\)
\(L(\frac12)\)  \(\approx\)  \(1.19867 - 1.25452i\)
\(L(2)\)   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 \( 1 + (-3.99 + 3.01i)T \)
7 \( 1 + (5.53 + 4.28i)T \)
good11 \( 1 + (-8.47 + 14.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-3.05 + 3.05i)T - 169iT^{2} \)
17 \( 1 + (-10.7 + 2.87i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-5.74 + 3.31i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (16.7 + 4.49i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 35.8iT - 841T^{2} \)
31 \( 1 + (-22.7 + 39.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (4.74 - 17.7i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 42.9T + 1.68e3T^{2} \)
43 \( 1 + (1.98 - 1.98i)T - 1.84e3iT^{2} \)
47 \( 1 + (11.3 - 42.2i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-22.0 - 82.1i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (22.0 + 12.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-42.4 - 73.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-60.8 + 16.2i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 137.T + 5.04e3T^{2} \)
73 \( 1 + (-1.87 - 7.00i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-35.4 + 20.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (109. - 109. i)T - 6.88e3iT^{2} \)
89 \( 1 + (-49.9 + 28.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (39.7 + 39.7i)T + 9.40e3iT^{2} \)
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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

−11.87523507091361747503438133198, −10.79465771840548825697276889516, −9.931473189252400147910553454629, −9.243452363405038163911643193473, −8.267596290316131220719155839507, −6.40006001429211661944882346093, −5.45671617005145398534225738882, −4.05996239338974115048351062293, −3.01104620453312689608257268273, −0.979287620844757080949829086384, 2.02898147469285633374841616308, 3.57527433828007208905172680487, 5.35548988323097417794575230690, 6.45474567685517835841174390117, 6.94618233376321694133092711042, 8.282318397836358266628303250671, 9.519708965119014342030231065233, 10.02133655800746838515868771878, 11.80103990826772877347003357481, 12.49752079793126378030850539688

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