Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.128 - 0.991i$
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 + (−4.98 + 0.371i)5-s − 2.44·6-s + (−6.80 − 1.63i)7-s + (2 + 1.99i)8-s + (−2.59 − 1.50i)9-s + (2.33 + 6.67i)10-s + (9.66 + 16.7i)11-s + (0.896 + 3.34i)12-s + (10.3 + 10.3i)13-s + (0.253 + 9.89i)14-s + (−1.61 + 8.50i)15-s + (1.99 − 3.46i)16-s + (−23.1 − 6.19i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.149 − 0.557i)3-s + (−0.433 + 0.250i)4-s + (−0.997 + 0.0743i)5-s − 0.408·6-s + (−0.972 − 0.234i)7-s + (0.250 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.233 + 0.667i)10-s + (0.879 + 1.52i)11-s + (0.0747 + 0.278i)12-s + (0.799 + 0.799i)13-s + (0.0180 + 0.706i)14-s + (−0.107 + 0.567i)15-s + (0.124 − 0.216i)16-s + (−1.35 − 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.128 - 0.991i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.128 - 0.991i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.255505 + 0.224632i\)
\(L(\frac12)\)  \(\approx\)  \(0.255505 + 0.224632i\)
\(L(2)\)   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 \( 1 + (4.98 - 0.371i)T \)
7 \( 1 + (6.80 + 1.63i)T \)
good11 \( 1 + (-9.66 - 16.7i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.3 - 10.3i)T + 169iT^{2} \)
17 \( 1 + (23.1 + 6.19i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (2.70 + 1.55i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (34.2 - 9.18i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 36.5iT - 841T^{2} \)
31 \( 1 + (1.49 + 2.59i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (0.113 + 0.422i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 64.3T + 1.68e3T^{2} \)
43 \( 1 + (38.9 + 38.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (2.88 + 10.7i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (7.95 - 29.6i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-34.4 + 19.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (21.4 - 37.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-3.98 - 1.06i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 81.1T + 5.04e3T^{2} \)
73 \( 1 + (-27.1 + 101. i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-25.8 - 14.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-6.76 - 6.76i)T + 6.88e3iT^{2} \)
89 \( 1 + (101. + 58.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (63.8 - 63.8i)T - 9.40e3iT^{2} \)
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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

−12.13944603039284194025180616887, −11.69398596545068837207506889892, −10.49330698629029504414184497491, −9.366919096867275089599032601484, −8.582308651239487704901311991778, −7.17730424416170783883080584308, −6.61647298693427694237358224220, −4.42730022733410687947500899012, −3.54774705725875164176038418864, −1.84093189062143835193952411122, 0.19685344721207137413332398144, 3.35313574788175569423283253044, 4.18858075828624672584248072068, 5.89134314831515591341091784256, 6.60379241432579344122708089832, 8.331843858940860206572955212162, 8.543717189803139003920957841588, 9.807961274232525457982954383038, 10.91291883441798517074761367185, 11.77991606893779988647519815664

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