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Properties

Label 2-210-7.3-c2-0-3
Degree $2$
Conductor $210$
Sign $-0.854 - 0.519i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.854 - 0.519i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.854 - 0.519i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.401004 + 1.43046i\)
\(L(\frac12)\) \(\approx\) \(0.401004 + 1.43046i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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