Properties

Label 2-210-5.3-c2-0-11
Degree $2$
Conductor $210$
Sign $-0.940 + 0.340i$
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 + (−0.578 − 4.96i)5-s − 2.44·6-s + (1.87 − 1.87i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−5.54 − 4.38i)10-s − 12.4·11-s + (−2.44 + 2.44i)12-s + (3.13 + 3.13i)13-s − 3.74i·14-s + (−5.37 + 6.79i)15-s − 4·16-s + (−5.80 + 5.80i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (−0.115 − 0.993i)5-s − 0.408·6-s + (0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.554 − 0.438i)10-s − 1.12·11-s + (−0.204 + 0.204i)12-s + (0.241 + 0.241i)13-s − 0.267i·14-s + (−0.358 + 0.452i)15-s − 0.250·16-s + (−0.341 + 0.341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.940 + 0.340i$
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: \((2,\ 210,\ (\ :1),\ -0.940 + 0.340i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.233664 - 1.33003i\)
\(L(\frac12)\) \(\approx\) \(0.233664 - 1.33003i\)
\(L(2)\) 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 \( 1 + (0.578 + 4.96i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good11 \( 1 + 12.4T + 121T^{2} \)
13 \( 1 + (-3.13 - 3.13i)T + 169iT^{2} \)
17 \( 1 + (5.80 - 5.80i)T - 289iT^{2} \)
19 \( 1 + 26.5iT - 361T^{2} \)
23 \( 1 + (10.4 + 10.4i)T + 529iT^{2} \)
29 \( 1 + 14.5iT - 841T^{2} \)
31 \( 1 - 42.6T + 961T^{2} \)
37 \( 1 + (-11.9 + 11.9i)T - 1.36e3iT^{2} \)
41 \( 1 - 37.5T + 1.68e3T^{2} \)
43 \( 1 + (-24.0 - 24.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-8.83 + 8.83i)T - 2.20e3iT^{2} \)
53 \( 1 + (1.97 + 1.97i)T + 2.80e3iT^{2} \)
59 \( 1 + 88.2iT - 3.48e3T^{2} \)
61 \( 1 - 102.T + 3.72e3T^{2} \)
67 \( 1 + (22.8 - 22.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 10.7T + 5.04e3T^{2} \)
73 \( 1 + (-80.4 - 80.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 138. iT - 6.24e3T^{2} \)
83 \( 1 + (96.0 + 96.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 3.29iT - 7.92e3T^{2} \)
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

−11.74192605907796262559850200872, −11.02648067032715874509628690227, −9.958133741588331880489607249931, −8.697671635365358027899623936137, −7.69655047408575525969710040052, −6.29501124752707469534053042126, −5.12015398653510732720568641796, −4.30641664752861511289395646882, −2.36386048854697758808198580105, −0.67161952533483485419618285373, 2.69715732058352496712588823737, 4.01753136566828914490718200427, 5.37684876023716186632437867365, 6.20497961381797694874035803457, 7.44063405935776294592685769891, 8.309446717150341544182982181754, 9.875877217940338713877134116025, 10.69514067458704460754521630786, 11.60432787351453706019424424804, 12.53930150990362718083491856400

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