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Properties

Label 2-210-5.3-c2-0-10
Degree $2$
Conductor $210$
Sign $-0.985 + 0.172i$
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  + (−1 + i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + (0.294 − 4.99i)5-s + 2.44·6-s + (−1.87 + 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s + (4.69 + 5.28i)10-s − 3.32·11-s + (−2.44 + 2.44i)12-s + (−7.80 − 7.80i)13-s − 3.74i·14-s + (−6.47 + 5.75i)15-s − 4·16-s + (−13.6 + 13.6i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (0.0588 − 0.998i)5-s + 0.408·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.469 + 0.528i)10-s − 0.302·11-s + (−0.204 + 0.204i)12-s + (−0.600 − 0.600i)13-s − 0.267i·14-s + (−0.431 + 0.383i)15-s − 0.250·16-s + (−0.803 + 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.985 + 0.172i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.985 + 0.172i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0146998 - 0.169528i\)
\(L(\frac12)\) \(\approx\) \(0.0146998 - 0.169528i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (-0.294 + 4.99i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good11 \( 1 + 3.32T + 121T^{2} \)
13 \( 1 + (7.80 + 7.80i)T + 169iT^{2} \)
17 \( 1 + (13.6 - 13.6i)T - 289iT^{2} \)
19 \( 1 - 37.1iT - 361T^{2} \)
23 \( 1 + (31.5 + 31.5i)T + 529iT^{2} \)
29 \( 1 + 55.3iT - 841T^{2} \)
31 \( 1 + 4.54T + 961T^{2} \)
37 \( 1 + (46.8 - 46.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 9.94T + 1.68e3T^{2} \)
43 \( 1 + (-23.0 - 23.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (6.38 - 6.38i)T - 2.20e3iT^{2} \)
53 \( 1 + (42.9 + 42.9i)T + 2.80e3iT^{2} \)
59 \( 1 - 59.2iT - 3.48e3T^{2} \)
61 \( 1 - 47.1T + 3.72e3T^{2} \)
67 \( 1 + (-8.48 + 8.48i)T - 4.48e3iT^{2} \)
71 \( 1 - 85.6T + 5.04e3T^{2} \)
73 \( 1 + (34.7 + 34.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 96.4iT - 6.24e3T^{2} \)
83 \( 1 + (-19.6 - 19.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 43.0iT - 7.92e3T^{2} \)
97 \( 1 + (-88.5 + 88.5i)T - 9.40e3iT^{2} \)
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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

−11.93661983413655836726384424856, −10.42182381239695733918591096018, −9.760345671233017028737432153721, −8.321231654370718505352538128609, −7.974892879209196487596374541305, −6.34836365186081360856820187865, −5.65787118177146146358832549994, −4.33935082863977847939364720976, −1.94338449547543272604344697677, −0.11041933225371415237369802690, 2.38338886041956038850265206394, 3.69762939800177618409605046022, 5.13073628416960392742624461572, 6.76919383194307107435790970221, 7.35315330599692273850366052687, 9.058495831365641138116432958316, 9.727548388293960920618480245967, 10.82720872705099695181334950124, 11.25669395007928178459719419703, 12.30487847462329057646483482011

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