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Properties

Label 2-5292-63.41-c1-0-38
Degree $2$
Conductor $5292$
Sign $-0.979 + 0.203i$
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  + (1.95 − 3.39i)5-s + (−3.19 + 1.84i)11-s + (0.480 + 0.277i)13-s + 5.83·17-s − 5.33i·19-s + (−1.96 − 1.13i)23-s + (−5.16 − 8.94i)25-s + (−3.53 + 2.04i)29-s + (−7.00 − 4.04i)31-s − 7.79·37-s + (−3.59 + 6.22i)41-s + (−0.754 − 1.30i)43-s + (−1.41 − 2.44i)47-s + 0.0479i·53-s + 14.4i·55-s + ⋯
L(s)  = 1  + (0.875 − 1.51i)5-s + (−0.964 + 0.556i)11-s + (0.133 + 0.0769i)13-s + 1.41·17-s − 1.22i·19-s + (−0.410 − 0.237i)23-s + (−1.03 − 1.78i)25-s + (−0.656 + 0.379i)29-s + (−1.25 − 0.726i)31-s − 1.28·37-s + (−0.561 + 0.971i)41-s + (−0.114 − 0.199i)43-s + (−0.206 − 0.357i)47-s + 0.00659i·53-s + 1.95i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.979 + 0.203i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.979 + 0.203i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.153324936\)
\(L(\frac12)\) \(\approx\) \(1.153324936\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 + 3.39i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.19 - 1.84i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.480 - 0.277i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.83T + 17T^{2} \)
19 \( 1 + 5.33iT - 19T^{2} \)
23 \( 1 + (1.96 + 1.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.53 - 2.04i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.00 + 4.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.79T + 37T^{2} \)
41 \( 1 + (3.59 - 6.22i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.754 + 1.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.0479iT - 53T^{2} \)
59 \( 1 + (4.45 - 7.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.03 + 3.48i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.587 - 1.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.71iT - 71T^{2} \)
73 \( 1 + 4.07iT - 73T^{2} \)
79 \( 1 + (-1.97 - 3.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.84 + 6.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.42T + 89T^{2} \)
97 \( 1 + (-13.9 + 8.07i)T + (48.5 - 84.0i)T^{2} \)
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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

−7.87806230154031744953611570176, −7.30215067512401187243390214095, −6.25892579246807975697760678280, −5.37486843829960676178806837476, −5.18715146996129074425493347914, −4.36312067577595951456113925559, −3.28745920974603764069004805041, −2.17188915813707567198504980865, −1.45949685525390296558000686214, −0.27711334131123655559030853326, 1.56511280941431577833002798221, 2.36235817297237100477092521429, 3.38980198314570765582694078452, 3.59662549531580093772263439978, 5.33009245327070514985424628907, 5.59918919297133160826039733657, 6.29931865246379080865923403066, 7.15083271809080814747665301229, 7.68459959585225140723619252496, 8.383650252122834995613377651814

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