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Properties

Label 2-210-35.23-c2-0-4
Degree $2$
Conductor $210$
Sign $-0.0455 - 0.998i$
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  + (0.366 + 1.36i)2-s + (0.448 − 1.67i)3-s + (−1.73 + i)4-s + (3.99 + 3.01i)5-s + 2.44·6-s + (−5.53 + 4.28i)7-s + (−2 − 1.99i)8-s + (−2.59 − 1.50i)9-s + (−2.65 + 6.55i)10-s + (8.47 + 14.6i)11-s + (0.896 + 3.34i)12-s + (3.05 + 3.05i)13-s + (−7.87 − 5.99i)14-s + (6.82 − 5.32i)15-s + (1.99 − 3.46i)16-s + (10.7 + 2.87i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.149 − 0.557i)3-s + (−0.433 + 0.250i)4-s + (0.798 + 0.602i)5-s + 0.408·6-s + (−0.790 + 0.612i)7-s + (−0.250 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.265 + 0.655i)10-s + (0.770 + 1.33i)11-s + (0.0747 + 0.278i)12-s + (0.234 + 0.234i)13-s + (−0.562 − 0.428i)14-s + (0.455 − 0.355i)15-s + (0.124 − 0.216i)16-s + (0.632 + 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0455 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0455 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.0455 - 0.998i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.0455 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19867 + 1.25452i\)
\(L(\frac12)\) \(\approx\) \(1.19867 + 1.25452i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 + (-0.448 + 1.67i)T \)
5 \( 1 + (-3.99 - 3.01i)T \)
7 \( 1 + (5.53 - 4.28i)T \)
good11 \( 1 + (-8.47 - 14.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.05 - 3.05i)T + 169iT^{2} \)
17 \( 1 + (-10.7 - 2.87i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (-5.74 - 3.31i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (16.7 - 4.49i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 35.8iT - 841T^{2} \)
31 \( 1 + (-22.7 - 39.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (4.74 + 17.7i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 42.9T + 1.68e3T^{2} \)
43 \( 1 + (1.98 + 1.98i)T + 1.84e3iT^{2} \)
47 \( 1 + (11.3 + 42.2i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-22.0 + 82.1i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (22.0 - 12.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-42.4 + 73.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-60.8 - 16.2i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 137.T + 5.04e3T^{2} \)
73 \( 1 + (-1.87 + 7.00i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-35.4 - 20.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (109. + 109. i)T + 6.88e3iT^{2} \)
89 \( 1 + (-49.9 - 28.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (39.7 - 39.7i)T - 9.40e3iT^{2} \)
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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

−12.49752079793126378030850539688, −11.80103990826772877347003357481, −10.02133655800746838515868771878, −9.519708965119014342030231065233, −8.282318397836358266628303250671, −6.94618233376321694133092711042, −6.45474567685517835841174390117, −5.35548988323097417794575230690, −3.57527433828007208905172680487, −2.02898147469285633374841616308, 0.979287620844757080949829086384, 3.01104620453312689608257268273, 4.05996239338974115048351062293, 5.45671617005145398534225738882, 6.40006001429211661944882346093, 8.267596290316131220719155839507, 9.243452363405038163911643193473, 9.931473189252400147910553454629, 10.79465771840548825697276889516, 11.87523507091361747503438133198

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