Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.854 + 0.519i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (−1.93 − 1.11i)5-s − 2.44i·6-s + (−6.51 − 2.55i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (−2.73 + 1.58i)10-s + (−6.16 − 10.6i)11-s + (−2.99 − 1.73i)12-s − 7.26i·13-s + (−7.73 + 6.17i)14-s − 3.87·15-s + (−2.00 + 3.46i)16-s + (8.04 − 4.64i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s − 0.408i·6-s + (−0.930 − 0.365i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.273 + 0.158i)10-s + (−0.560 − 0.970i)11-s + (−0.249 − 0.144i)12-s − 0.558i·13-s + (−0.552 + 0.440i)14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (0.473 − 0.273i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.854 + 0.519i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (61, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.854 + 0.519i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.401004 - 1.43046i\)
\(L(\frac12)\)  \(\approx\)  \(0.401004 - 1.43046i\)
\(L(2)\)   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 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (6.51 + 2.55i)T \)
good11 \( 1 + (6.16 + 10.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 7.26iT - 169T^{2} \)
17 \( 1 + (-8.04 + 4.64i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-5.26 - 3.03i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-1.12 + 1.94i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 42.2T + 841T^{2} \)
31 \( 1 + (1.05 - 0.609i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (17.5 - 30.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 57.8iT - 1.68e3T^{2} \)
43 \( 1 + 34.0T + 1.84e3T^{2} \)
47 \( 1 + (-49.4 - 28.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (7.27 + 12.5i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-50.1 + 28.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (5.07 + 2.93i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (24.7 + 42.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 101.T + 5.04e3T^{2} \)
73 \( 1 + (-71.2 + 41.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (55.8 - 96.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 91.6iT - 6.88e3T^{2} \)
89 \( 1 + (-110. - 63.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 61.4iT - 9.40e3T^{2} \)
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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

−11.97931019107720821085395154391, −10.72847337918741028394263759013, −9.946403110571269977046366819277, −8.790860916409297294642129199407, −7.82509409913939593812070027169, −6.53135830042228563027661036660, −5.25259459094115082967270751852, −3.68229010343811104377055716010, −2.84968329835101944986895668365, −0.70644881144069948566373343580, 2.67381194644828292537135172196, 3.89935878444923605713000886653, 5.11578176935136806767891379815, 6.50597358880454369877769793829, 7.40518614228557921450496911172, 8.478148331149859390203542957954, 9.531713707032195399466420364456, 10.35878330684442847396072707607, 11.86797216018345457668512832811, 12.65000702228719115507991864552

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