Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.809 + 0.586i$
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 + (−4.97 + 0.538i)5-s − 2.44·6-s + (3.26 + 6.19i)7-s + (−2 − 1.99i)8-s + (−2.59 − 1.50i)9-s + (−2.55 − 6.59i)10-s + (1.81 + 3.15i)11-s + (−0.896 − 3.34i)12-s + (−14.2 − 14.2i)13-s + (−7.26 + 6.72i)14-s + (1.32 − 8.55i)15-s + (1.99 − 3.46i)16-s + (−20.6 − 5.53i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.149 + 0.557i)3-s + (−0.433 + 0.250i)4-s + (−0.994 + 0.107i)5-s − 0.408·6-s + (0.466 + 0.884i)7-s + (−0.250 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.255 − 0.659i)10-s + (0.165 + 0.286i)11-s + (−0.0747 − 0.278i)12-s + (−1.09 − 1.09i)13-s + (−0.518 + 0.480i)14-s + (0.0884 − 0.570i)15-s + (0.124 − 0.216i)16-s + (−1.21 − 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.809 + 0.586i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.809 + 0.586i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.175873 - 0.542524i\)
\(L(\frac12)\)  \(\approx\)  \(0.175873 - 0.542524i\)
\(L(2)\)   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 \( 1 + (4.97 - 0.538i)T \)
7 \( 1 + (-3.26 - 6.19i)T \)
good11 \( 1 + (-1.81 - 3.15i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (14.2 + 14.2i)T + 169iT^{2} \)
17 \( 1 + (20.6 + 5.53i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (-0.949 - 0.548i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (24.3 - 6.52i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 1.99iT - 841T^{2} \)
31 \( 1 + (-25.3 - 43.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-11.8 - 44.3i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 18.6T + 1.68e3T^{2} \)
43 \( 1 + (6.49 + 6.49i)T + 1.84e3iT^{2} \)
47 \( 1 + (0.305 + 1.13i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (24.4 - 91.2i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (95.2 - 54.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-55.8 + 96.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-11.9 - 3.19i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 69.2T + 5.04e3T^{2} \)
73 \( 1 + (12.4 - 46.3i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-64.8 - 37.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (28.7 + 28.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (-24.1 - 13.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (101. - 101. i)T - 9.40e3iT^{2} \)
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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

−12.44593912623528031695952832967, −11.95618309035897850814282989892, −10.84217512194605701130725231023, −9.667218064797167859697361134490, −8.542457606806940460102488201755, −7.79792672572064105407939784983, −6.58548786334393699113345410640, −5.20353441570323490521719733000, −4.46123352413497678602103873603, −2.92179933773179955175399018240, 0.29764482132771626627806136866, 2.10313209919163577027088592422, 3.95594227114856578345724125310, 4.70427081448894990885076963863, 6.50637528730472939753613240250, 7.54147262567166425243928252230, 8.470417835289351920146863359628, 9.735663205169454507230057613800, 11.00169358704652265984720708653, 11.54841654776716829614046112991

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