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Properties

Label 2-5292-9.4-c1-0-16
Degree $2$
Conductor $5292$
Sign $-0.173 - 0.984i$
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  + (−1.5 + 2.59i)5-s + (1.5 + 2.59i)11-s + (−0.5 + 0.866i)13-s + 6·17-s + 4·19-s + (−1.5 + 2.59i)23-s + (−2 − 3.46i)25-s + (1.5 + 2.59i)29-s + (2.5 − 4.33i)31-s + 2·37-s + (−1.5 + 2.59i)41-s + (0.5 + 0.866i)43-s + (4.5 + 7.79i)47-s + 6·53-s − 9·55-s + ⋯
L(s)  = 1  + (−0.670 + 1.16i)5-s + (0.452 + 0.783i)11-s + (−0.138 + 0.240i)13-s + 1.45·17-s + 0.917·19-s + (−0.312 + 0.541i)23-s + (−0.400 − 0.692i)25-s + (0.278 + 0.482i)29-s + (0.449 − 0.777i)31-s + 0.328·37-s + (−0.234 + 0.405i)41-s + (0.0762 + 0.132i)43-s + (0.656 + 1.13i)47-s + 0.824·53-s − 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (1765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.821892363\)
\(L(\frac12)\) \(\approx\) \(1.821892363\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-5.5 - 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−7.932039095754783341348644935743, −7.79541603452542223006465933987, −6.99457691377863456282227520066, −6.42536399995803981029179137949, −5.52876379069087807078023722543, −4.67577952092098257811519411321, −3.72159708044528957798997012691, −3.24035441898331321716179609359, −2.28115010444354823183467351754, −1.08333234808291620716377415740, 0.62607065710655954729249065817, 1.23103351585545780779465793219, 2.69756767146503763468751867304, 3.62502513105806412459837991532, 4.20637727672175029854184478402, 5.23274886483859122615383116607, 5.56182083340702673658634069811, 6.57549982340142569083174315161, 7.46253276401390736968909661942, 8.105890648688986281629505052324

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