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Properties

Label 2-210-105.38-c2-0-24
Degree $2$
Conductor $210$
Sign $-0.0442 + 0.999i$
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 + (1.26 − 2.72i)3-s + (−1.73 − i)4-s + (−2.65 − 4.23i)5-s + (3.25 + 2.72i)6-s + (3.47 + 6.07i)7-s + (2 − 1.99i)8-s + (−5.80 − 6.87i)9-s + (6.75 − 2.07i)10-s + (−5.23 − 3.02i)11-s + (−4.90 + 3.45i)12-s + (7.33 − 7.33i)13-s + (−9.57 + 2.51i)14-s + (−14.8 + 1.87i)15-s + (1.99 + 3.46i)16-s + (−6.03 − 22.5i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.420 − 0.907i)3-s + (−0.433 − 0.250i)4-s + (−0.531 − 0.847i)5-s + (0.542 + 0.453i)6-s + (0.495 + 0.868i)7-s + (0.250 − 0.249i)8-s + (−0.645 − 0.763i)9-s + (0.675 − 0.207i)10-s + (−0.475 − 0.274i)11-s + (−0.409 + 0.287i)12-s + (0.564 − 0.564i)13-s + (−0.683 + 0.179i)14-s + (−0.992 + 0.125i)15-s + (0.124 + 0.216i)16-s + (−0.354 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0442 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0442 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.0442 + 0.999i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.0442 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.777590 - 0.812823i\)
\(L(\frac12)\) \(\approx\) \(0.777590 - 0.812823i\)
\(L(2)\) 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 \( 1 + (-1.26 + 2.72i)T \)
5 \( 1 + (2.65 + 4.23i)T \)
7 \( 1 + (-3.47 - 6.07i)T \)
good11 \( 1 + (5.23 + 3.02i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.33 + 7.33i)T - 169iT^{2} \)
17 \( 1 + (6.03 + 22.5i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (17.0 + 29.6i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (26.5 + 7.11i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 6.86T + 841T^{2} \)
31 \( 1 + (-38.9 - 22.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-34.1 - 9.14i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 18.2T + 1.68e3T^{2} \)
43 \( 1 + (-17.2 - 17.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-23.3 - 6.24i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (20.7 + 77.2i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-97.9 - 56.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-16.8 + 9.74i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.92 + 10.9i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 80.6iT - 5.04e3T^{2} \)
73 \( 1 + (0.519 + 1.93i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-73.1 + 42.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (4.56 + 4.56i)T + 6.88e3iT^{2} \)
89 \( 1 + (-21.5 + 12.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (38.4 + 38.4i)T + 9.40e3iT^{2} \)
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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

−12.03801024964136316284080436326, −11.18562474679814127332046709956, −9.417765446853783139894272104990, −8.464491079595353357732749971112, −8.156021611316018972918500010086, −6.89983520546636486271420035561, −5.71876520182105406620844903570, −4.60208342374388280323022349952, −2.60763169702270969178182255952, −0.64001498164451027849261898609, 2.17528086847195180213887907849, 3.87637139490439009853535463039, 4.19120941809219167491487190229, 6.13728396793408851224702048484, 7.84787867755413253485083627123, 8.308323532034878303725326163748, 9.844711263056963839572231240892, 10.53568953391507497056442971862, 11.03325145830579262415518263325, 12.12506705446738113852867193978

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