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Properties

Label 2-1260-140.39-c0-0-1
Degree $2$
Conductor $1260$
Sign $0.832 + 0.553i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 0.999·8-s + (0.499 − 0.866i)10-s − 0.999·14-s + (−0.5 − 0.866i)16-s − 0.999·20-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.499 + 0.866i)28-s + 29-s + (−0.499 + 0.866i)32-s + 0.999·35-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 0.999·8-s + (0.499 − 0.866i)10-s − 0.999·14-s + (−0.5 − 0.866i)16-s − 0.999·20-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.499 + 0.866i)28-s + 29-s + (−0.499 + 0.866i)32-s + 0.999·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9295915988\)
\(L(\frac12)\) \(\approx\) \(0.9295915988\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.04700884362416143524957849800, −9.208334238707471672239904157458, −8.263061141618735344424156152540, −7.40830812135367188343760527756, −6.82925252878520269144858618209, −5.51981119917014092880316900209, −4.40324891477538227470864901513, −3.48532056306265839853612014249, −2.50620156542947502343355588274, −1.32947016126302348961974758193, 1.24154158826693006562837466492, 2.49773641200938855230480166436, 4.41906793821484964309742847429, 5.02024759444730388166881514973, 5.87835983479194488984376770738, 6.52304395211380699632281831855, 7.75282947023654254684735516156, 8.373003335991646403874980638019, 9.058419528450012762441197264133, 9.591054549577520070570865376802

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