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Properties

Label 2-5292-9.7-c1-0-28
Degree $2$
Conductor $5292$
Sign $0.766 + 0.642i$
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  + (1 + 1.73i)5-s + (2 − 3.46i)11-s + (−1.5 − 2.59i)13-s − 7·17-s + 5·19-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s + (−0.5 + 0.866i)29-s + (1.5 + 2.59i)31-s + 11·37-s + (−4.5 − 7.79i)41-s + (−2.5 + 4.33i)43-s + (1.5 − 2.59i)47-s − 3·53-s + 7.99·55-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (0.603 − 1.04i)11-s + (−0.416 − 0.720i)13-s − 1.69·17-s + 1.14·19-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + (−0.0928 + 0.160i)29-s + (0.269 + 0.466i)31-s + 1.80·37-s + (−0.702 − 1.21i)41-s + (−0.381 + 0.660i)43-s + (0.218 − 0.378i)47-s − 0.412·53-s + 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (3529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.954241321\)
\(L(\frac12)\) \(\approx\) \(1.954241321\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 11T + 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.096432636143850653166488770206, −7.31149809979171135891637925314, −6.61706161940970894091614936721, −6.07806453427207003368585551338, −5.29031963061729056388367555075, −4.45572687579287553979953728423, −3.35273085562019715385227528332, −2.89157418228949035411097426309, −1.85237026015284267678644527330, −0.58134295081294809973308100194, 1.02388004701166418935355461429, 1.95555841327081148811512009705, 2.73767309853704406212777867966, 4.10420893926056183715639464175, 4.59909170594762317703112129073, 5.18347126176157555556413580739, 6.25410170534140642804682541463, 6.78832048669187998558108714226, 7.48606639060235296571686745126, 8.370797942546796386827833187161

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