Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.0140 - 0.999i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.366i)2-s + (2.23 + 1.99i)3-s + (1.73 + i)4-s + (−2.72 + 4.19i)5-s + (2.32 + 3.54i)6-s + (−5.35 − 4.50i)7-s + (1.99 + 2i)8-s + (1.03 + 8.94i)9-s + (−5.25 + 4.73i)10-s + (8.15 + 4.70i)11-s + (1.88 + 5.69i)12-s + (16.1 + 16.1i)13-s + (−5.66 − 8.11i)14-s + (−14.4 + 3.95i)15-s + (1.99 + 3.46i)16-s + (−9.03 + 2.42i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.746 + 0.665i)3-s + (0.433 + 0.250i)4-s + (−0.544 + 0.838i)5-s + (0.388 + 0.591i)6-s + (−0.765 − 0.643i)7-s + (0.249 + 0.250i)8-s + (0.114 + 0.993i)9-s + (−0.525 + 0.473i)10-s + (0.741 + 0.428i)11-s + (0.156 + 0.474i)12-s + (1.24 + 1.24i)13-s + (−0.404 − 0.579i)14-s + (−0.964 + 0.263i)15-s + (0.124 + 0.216i)16-s + (−0.531 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0140 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0140 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.0140 - 0.999i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.0140 - 0.999i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.80274 + 1.77755i\)
\(L(\frac12)\)  \(\approx\)  \(1.80274 + 1.77755i\)
\(L(2)\)   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 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 + (-2.23 - 1.99i)T \)
5 \( 1 + (2.72 - 4.19i)T \)
7 \( 1 + (5.35 + 4.50i)T \)
good11 \( 1 + (-8.15 - 4.70i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-16.1 - 16.1i)T + 169iT^{2} \)
17 \( 1 + (9.03 - 2.42i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (13.1 + 22.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-6.16 + 23.0i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 34.0T + 841T^{2} \)
31 \( 1 + (-2.88 - 1.66i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-12.3 + 46.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 3.31T + 1.68e3T^{2} \)
43 \( 1 + (-28.9 + 28.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (-0.830 + 3.09i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (56.7 - 15.2i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-33.5 - 19.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-80.5 + 46.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-3.66 + 0.981i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 26.4iT - 5.04e3T^{2} \)
73 \( 1 + (-27.3 + 7.31i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-38.5 + 22.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (70.6 - 70.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (118. - 68.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
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

−12.55542909136171157232206102767, −11.17756312460350574794472093896, −10.69977840505830696526426319231, −9.390570569954721212929265271195, −8.437416446840877587099033243449, −6.95763266196523520657853132770, −6.52056496287022360193392435448, −4.34013205185941903302116232927, −3.89723422203518011706943956717, −2.55874494811735096872146442552, 1.20877046701009854195768237321, 3.06609996491090010085798098442, 3.96283863917077686133410840562, 5.71911522443582613462578388846, 6.55422282348783553504365458764, 8.085328946719804106225795567883, 8.680768132096056080888565946456, 9.813999966583496644979641982179, 11.33730150967036452012340748470, 12.21018446611284457959443160756

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