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Properties

Label 2-5292-63.58-c1-0-28
Degree $2$
Conductor $5292$
Sign $0.248 + 0.968i$
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  + (−0.849 − 1.47i)5-s + (1.23 − 2.14i)11-s + (−0.388 + 0.673i)13-s + (1.40 + 2.43i)17-s + (2.49 − 4.31i)19-s + (0.356 + 0.616i)23-s + (1.05 − 1.82i)25-s + (2.25 + 3.90i)29-s + 5.09·31-s + (3.43 − 5.95i)37-s + (−2.93 + 5.08i)41-s + (2.32 + 4.03i)43-s − 12.9·47-s + (0.944 + 1.63i)53-s − 4.21·55-s + ⋯
L(s)  = 1  + (−0.380 − 0.658i)5-s + (0.373 − 0.646i)11-s + (−0.107 + 0.186i)13-s + (0.340 + 0.590i)17-s + (0.572 − 0.990i)19-s + (0.0742 + 0.128i)23-s + (0.211 − 0.365i)25-s + (0.418 + 0.725i)29-s + 0.915·31-s + (0.565 − 0.979i)37-s + (−0.458 + 0.794i)41-s + (0.354 + 0.614i)43-s − 1.89·47-s + (0.129 + 0.224i)53-s − 0.567·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.248 + 0.968i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.248 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.747529972\)
\(L(\frac12)\) \(\approx\) \(1.747529972\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.849 + 1.47i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.23 + 2.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.388 - 0.673i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.40 - 2.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.49 + 4.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.356 - 0.616i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.25 - 3.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.09T + 31T^{2} \)
37 \( 1 + (-3.43 + 5.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.93 - 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.32 - 4.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + (-0.944 - 1.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 - 7.98T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + (2.49 + 4.31i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 9.21T + 79T^{2} \)
83 \( 1 + (-4.40 - 7.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.82 + 8.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.32 + 7.48i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.323416670203875161766942832509, −7.35001964678731449747601182381, −6.59277360287008892834274304748, −5.91786769229116983784517303676, −4.96964559213877073102379370796, −4.48229752438647560016376901586, −3.51274990064895148689019083429, −2.77458965192279035185058904402, −1.47247214374992352095590582645, −0.57299201790454614068121462403, 0.996608285803969724660504649358, 2.17160185799348920893442883756, 3.11558385049486037534245011319, 3.75133174560591021862597504624, 4.70293477988011667947360682972, 5.38172228649722749065770203542, 6.40096523219465246423304366817, 6.85802860668670677014709657084, 7.70677061241452662214578218253, 8.102189082481040155822148469350

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