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Properties

Label 2-1014-13.12-c3-0-57
Degree $2$
Conductor $1014$
Sign $0.0304 + 0.999i$
Analytic cond. $59.8279$
Root an. cond. $7.73485$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 3·3-s − 4·4-s + 17.9i·5-s − 6i·6-s − 26.1i·7-s + 8i·8-s + 9·9-s + 35.8·10-s − 12.4i·11-s − 12·12-s − 52.3·14-s + 53.7i·15-s + 16·16-s + 115.·17-s − 18i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 1.60i·5-s − 0.408i·6-s − 1.41i·7-s + 0.353i·8-s + 0.333·9-s + 1.13·10-s − 0.341i·11-s − 0.288·12-s − 0.998·14-s + 0.925i·15-s + 0.250·16-s + 1.65·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.0304 + 0.999i$
Analytic conductor: \(59.8279\)
Root analytic conductor: \(7.73485\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :3/2),\ 0.0304 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.249797212\)
\(L(\frac12)\) \(\approx\) \(2.249797212\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 - 3T \)
13 \( 1 \)
good5 \( 1 - 17.9iT - 125T^{2} \)
7 \( 1 + 26.1iT - 343T^{2} \)
11 \( 1 + 12.4iT - 1.33e3T^{2} \)
17 \( 1 - 115.T + 4.91e3T^{2} \)
19 \( 1 + 92.5iT - 6.85e3T^{2} \)
23 \( 1 + 203.T + 1.21e4T^{2} \)
29 \( 1 - 165.T + 2.43e4T^{2} \)
31 \( 1 - 223. iT - 2.97e4T^{2} \)
37 \( 1 - 137. iT - 5.06e4T^{2} \)
41 \( 1 + 371. iT - 6.89e4T^{2} \)
43 \( 1 + 131.T + 7.95e4T^{2} \)
47 \( 1 + 82.8iT - 1.03e5T^{2} \)
53 \( 1 - 433.T + 1.48e5T^{2} \)
59 \( 1 + 459. iT - 2.05e5T^{2} \)
61 \( 1 + 376.T + 2.26e5T^{2} \)
67 \( 1 + 206. iT - 3.00e5T^{2} \)
71 \( 1 + 264. iT - 3.57e5T^{2} \)
73 \( 1 + 843. iT - 3.89e5T^{2} \)
79 \( 1 - 869.T + 4.93e5T^{2} \)
83 \( 1 + 879. iT - 5.71e5T^{2} \)
89 \( 1 + 161. iT - 7.04e5T^{2} \)
97 \( 1 + 1.66e3iT - 9.12e5T^{2} \)
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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

−9.808596721908402350981161282707, −8.519976208960254134423339528731, −7.61879643027433494528875246108, −7.06157924758405817176968014835, −6.10520963573439240225794816972, −4.65577039281714781405091714211, −3.48023596157635018200273263813, −3.24218369336003937451128786384, −1.97070244181056074148774072070, −0.58715826825237784827102308924, 1.10000053046865371486938872200, 2.24580436238005407946909475114, 3.74116399258555937922014309809, 4.67530996053803612717857691726, 5.63927941999061428271905331827, 6.01977933710932598156204949161, 7.69014273725249303789755431698, 8.190340866911228174635428721486, 8.702071620331267409853204397991, 9.685736893391768400361633438804

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