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Properties

Label 2-5292-63.38-c1-0-19
Degree $2$
Conductor $5292$
Sign $0.838 + 0.545i$
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  + (0.0382 − 0.0661i)5-s + (−4.66 + 2.69i)11-s + (−4.60 + 2.65i)13-s + (1.89 − 3.27i)17-s + (4.33 − 2.50i)19-s + (2.02 + 1.16i)23-s + (2.49 + 4.32i)25-s + (−8.84 − 5.10i)29-s − 5.74i·31-s + (0.354 + 0.613i)37-s + (−3.29 − 5.71i)41-s + (0.716 − 1.24i)43-s − 2.92·47-s + (10.4 + 6.05i)53-s + 0.411i·55-s + ⋯
L(s)  = 1  + (0.0170 − 0.0295i)5-s + (−1.40 + 0.811i)11-s + (−1.27 + 0.737i)13-s + (0.458 − 0.794i)17-s + (0.995 − 0.574i)19-s + (0.422 + 0.243i)23-s + (0.499 + 0.865i)25-s + (−1.64 − 0.948i)29-s − 1.03i·31-s + (0.0582 + 0.100i)37-s + (−0.515 − 0.892i)41-s + (0.109 − 0.189i)43-s − 0.426·47-s + (1.44 + 0.831i)53-s + 0.0554i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.838 + 0.545i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.838 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.318972455\)
\(L(\frac12)\) \(\approx\) \(1.318972455\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0382 + 0.0661i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.66 - 2.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.60 - 2.65i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.89 + 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.33 + 2.50i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.02 - 1.16i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.84 + 5.10i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.74iT - 31T^{2} \)
37 \( 1 + (-0.354 - 0.613i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.29 + 5.71i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.716 + 1.24i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.92T + 47T^{2} \)
53 \( 1 + (-10.4 - 6.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.579T + 59T^{2} \)
61 \( 1 + 2.77iT - 61T^{2} \)
67 \( 1 - 5.27T + 67T^{2} \)
71 \( 1 - 3.32iT - 71T^{2} \)
73 \( 1 + (-6.17 - 3.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.938T + 79T^{2} \)
83 \( 1 + (-6.49 + 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.51 - 2.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.18 - 3.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−7.85001580745665304876394348203, −7.29193440380697260618784038077, −7.13000052494649208008925271497, −5.74529324229534795643233792594, −5.15116557516018089213835335632, −4.67732074418578700521020465120, −3.57585279702263793331771886308, −2.59861132251097911437283151401, −2.02051361837453416934749806933, −0.47759105023215146941205574218, 0.74976603137109482935398262669, 2.07539062937666580182782357214, 3.03710970928355293928691878861, 3.50926672236250170845988809635, 4.88767036830091786048360226507, 5.28885011050758736972417036390, 5.93412899315245152673515068634, 6.96564182570229431064325156269, 7.64537370806933564262326493812, 8.142809088179220565215126720938

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