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Properties

Label 2-210-21.5-c1-0-9
Degree $2$
Conductor $210$
Sign $0.981 - 0.189i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (1.12 − 1.31i)3-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (1.63 − 0.581i)6-s + (1.26 + 2.32i)7-s + 0.999i·8-s + (−0.481 − 2.96i)9-s + (−0.866 + 0.499i)10-s + (0.834 − 0.481i)11-s + (1.70 + 0.312i)12-s − 1.98i·13-s + (−0.0658 + 2.64i)14-s + (0.581 + 1.63i)15-s + (−0.5 + 0.866i)16-s + (−1.31 − 2.27i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.647 − 0.761i)3-s + (0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.666 − 0.237i)6-s + (0.478 + 0.878i)7-s + 0.353i·8-s + (−0.160 − 0.987i)9-s + (−0.273 + 0.158i)10-s + (0.251 − 0.145i)11-s + (0.491 + 0.0900i)12-s − 0.551i·13-s + (−0.0175 + 0.706i)14-s + (0.150 + 0.421i)15-s + (−0.125 + 0.216i)16-s + (−0.319 − 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.981 - 0.189i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.981 - 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93301 + 0.184976i\)
\(L(\frac12)\) \(\approx\) \(1.93301 + 0.184976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.12 + 1.31i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-1.26 - 2.32i)T \)
good11 \( 1 + (-0.834 + 0.481i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.98iT - 13T^{2} \)
17 \( 1 + (1.31 + 2.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.47 + 2.58i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.33 + 1.34i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.90iT - 29T^{2} \)
31 \( 1 + (2.96 - 1.71i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.57 - 7.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.12T + 41T^{2} \)
43 \( 1 - 1.54T + 43T^{2} \)
47 \( 1 + (-5.19 + 8.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.4 + 6.58i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.05 - 1.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.76 + 4.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.897 - 1.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + (-10.1 + 5.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.204 + 0.353i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.53T + 83T^{2} \)
89 \( 1 + (5.76 - 9.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.6iT - 97T^{2} \)
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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

−12.46724427313225892985594919150, −11.80967220926878853732842503403, −10.68958232711291487446328006852, −8.984536500934592459217555989444, −8.357155640327075236294560759576, −7.21088890484667326691461982505, −6.36241992490577441490234314334, −5.06697215433143545619265112701, −3.44063029255916390514281001303, −2.23944288978295788039777359274, 2.05963177425062790591104038444, 4.03770767977534519535196429728, 4.25130730865024983351246373585, 5.81521251684333093887628179312, 7.40189185205069140055371839032, 8.436839360561081188601899638496, 9.540685569294650967488439652749, 10.52046466074257946882195725250, 11.25112414257034884415000789285, 12.40250930084165639485517688365

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