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Properties

Label 2-1210-5.4-c1-0-47
Degree $2$
Conductor $1210$
Sign $-0.998 - 0.0599i$
Analytic cond. $9.66189$
Root an. cond. $3.10835$
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  − i·2-s − 0.732i·3-s − 4-s + (0.133 − 2.23i)5-s − 0.732·6-s − 1.26i·7-s + i·8-s + 2.46·9-s + (−2.23 − 0.133i)10-s + 0.732i·12-s − 2.46i·13-s − 1.26·14-s + (−1.63 − 0.0980i)15-s + 16-s − 1.73i·17-s − 2.46i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.422i·3-s − 0.5·4-s + (0.0599 − 0.998i)5-s − 0.298·6-s − 0.479i·7-s + 0.353i·8-s + 0.821·9-s + (−0.705 − 0.0423i)10-s + 0.211i·12-s − 0.683i·13-s − 0.338·14-s + (−0.421 − 0.0253i)15-s + 0.250·16-s − 0.420i·17-s − 0.580i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1210\)    =    \(2 \cdot 5 \cdot 11^{2}\)
Sign: $-0.998 - 0.0599i$
Analytic conductor: \(9.66189\)
Root analytic conductor: \(3.10835\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1210} (969, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1210,\ (\ :1/2),\ -0.998 - 0.0599i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.391832921\)
\(L(\frac12)\) \(\approx\) \(1.391832921\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-0.133 + 2.23i)T \)
11 \( 1 \)
good3 \( 1 + 0.732iT - 3T^{2} \)
7 \( 1 + 1.26iT - 7T^{2} \)
13 \( 1 + 2.46iT - 13T^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
19 \( 1 + 4.19T + 19T^{2} \)
23 \( 1 + 2.73iT - 23T^{2} \)
29 \( 1 - 3.73T + 29T^{2} \)
31 \( 1 - 8.73T + 31T^{2} \)
37 \( 1 + 7.92iT - 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 9.46iT - 43T^{2} \)
47 \( 1 - 6.73iT - 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 - 7.26iT - 67T^{2} \)
71 \( 1 - 6.92T + 71T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 + 7.26T + 79T^{2} \)
83 \( 1 + 3.26iT - 83T^{2} \)
89 \( 1 + 0.464T + 89T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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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

−9.472032853306490707638687914730, −8.465617862919761411627417147630, −7.932725059461188403098934706055, −6.88116336403949816483449060723, −5.89277022493972049193669114217, −4.66278982559386501864246474235, −4.27660200502225811263495481819, −2.86528595672386139832895765800, −1.60942356754240932697359212442, −0.61903816890535063036048020988, 1.87178310465858464815611250719, 3.23696529384178593008017145244, 4.21560592129794576683222361227, 5.04735686081203949448755454825, 6.30487904974279642651275116295, 6.64518721559147837664269749543, 7.60497810466701758136988024958, 8.499087388036679439211714951622, 9.306110326255929237657207851467, 10.22896006292381987378300503592

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