Properties

Label 2-5292-63.4-c1-0-10
Degree $2$
Conductor $5292$
Sign $0.702 + 0.711i$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
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  − 2.46·5-s − 4.64·11-s + (−3.55 + 6.15i)13-s + (−2.25 + 3.90i)17-s + (−2.16 − 3.74i)19-s − 5.86·23-s + 1.05·25-s + (−3.48 − 6.04i)29-s + (3.69 + 6.39i)31-s + (0.363 + 0.629i)37-s + (0.136 − 0.236i)41-s + (2.41 + 4.18i)43-s + (1.83 − 3.18i)47-s + (2.52 − 4.37i)53-s + 11.4·55-s + ⋯
L(s)  = 1  − 1.10·5-s − 1.40·11-s + (−0.985 + 1.70i)13-s + (−0.547 + 0.948i)17-s + (−0.496 − 0.859i)19-s − 1.22·23-s + 0.210·25-s + (−0.647 − 1.12i)29-s + (0.662 + 1.14i)31-s + (0.0597 + 0.103i)37-s + (0.0213 − 0.0369i)41-s + (0.368 + 0.638i)43-s + (0.267 − 0.463i)47-s + (0.347 − 0.601i)53-s + 1.54·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3650153810\)
\(L(\frac12)\) \(\approx\) \(0.3650153810\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.46T + 5T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
13 \( 1 + (3.55 - 6.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.25 - 3.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.16 + 3.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.86T + 23T^{2} \)
29 \( 1 + (3.48 + 6.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.69 - 6.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.363 - 0.629i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.136 + 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.83 + 3.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.52 + 4.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.56 - 7.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.90 + 11.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.663 - 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + (-2.16 + 3.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.21 - 5.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.742 - 1.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.91 - 8.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.246 - 0.426i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.131976864517294425558332578219, −7.40841049021429004784001109863, −6.82400149887855878920163780510, −6.02269242463879406422697718171, −4.97340349001216928697853997992, −4.35941021195617747660289243569, −3.82733899648690135246109846404, −2.58120877366627035255740001180, −1.97660204511248777004726491455, −0.18628751706720502297019825778, 0.49739671117448717280588803998, 2.26514394622200435146606102671, 2.90710366775551102757480493694, 3.81814045795120704091102484720, 4.60751435724671358036984985003, 5.39251706389357375487103639178, 5.93732221541432454582743124891, 7.22318962416584082338763646029, 7.64967909792760413702112915343, 8.037457190333819010922139867535

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