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Properties

Label 2-5292-9.7-c1-0-15
Degree $2$
Conductor $5292$
Sign $0.969 + 0.244i$
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  + (−1.73 − 3.01i)5-s + (1.25 − 2.17i)11-s + (0.292 + 0.505i)13-s + 1.09·17-s + 5.93·19-s + (3.19 + 5.52i)23-s + (−3.55 + 6.15i)25-s + (−0.918 + 1.59i)29-s + (3.51 + 6.09i)31-s − 1.40·37-s + (5.37 + 9.31i)41-s + (−5.67 + 9.83i)43-s + (3.76 − 6.52i)47-s − 11.6·53-s − 8.75·55-s + ⋯
L(s)  = 1  + (−0.778 − 1.34i)5-s + (0.379 − 0.656i)11-s + (0.0810 + 0.140i)13-s + 0.265·17-s + 1.36·19-s + (0.665 + 1.15i)23-s + (−0.710 + 1.23i)25-s + (−0.170 + 0.295i)29-s + (0.631 + 1.09i)31-s − 0.231·37-s + (0.839 + 1.45i)41-s + (−0.866 + 1.49i)43-s + (0.549 − 0.951i)47-s − 1.59·53-s − 1.18·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.969 + 0.244i$
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: \((2,\ 5292,\ (\ :1/2),\ 0.969 + 0.244i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.725427992\)
\(L(\frac12)\) \(\approx\) \(1.725427992\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 3.01i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.25 + 2.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.292 - 0.505i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.09T + 17T^{2} \)
19 \( 1 - 5.93T + 19T^{2} \)
23 \( 1 + (-3.19 - 5.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.918 - 1.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.51 - 6.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.40T + 37T^{2} \)
41 \( 1 + (-5.37 - 9.31i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.67 - 9.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.76 + 6.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (-2.22 - 3.85i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.17 + 10.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.33 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.93T + 71T^{2} \)
73 \( 1 + 8.71T + 73T^{2} \)
79 \( 1 + (-0.280 + 0.485i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.68 - 6.38i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-6.98 + 12.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.205247200058988239575264067002, −7.61857223856894625428515142042, −6.82638071869257765277211179790, −5.86047876103917422879675963781, −5.10850069150856289730395508676, −4.64136459623201484486874842300, −3.60197143689819882803956874266, −3.10434907233404034687523148238, −1.43182770628253454251193987165, −0.887879554770390726181768170431, 0.63862241610458987374233357384, 2.12737213023705585827961442597, 2.95750819363054034324212895485, 3.65449837244439817119419708065, 4.38903113300887272663000955577, 5.33456931494079254811261209996, 6.25592527323224329597030043978, 6.90640342395877944702471338219, 7.46036376134400603165335283666, 7.959473890313042684127607374617

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