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Properties

Label 2-5292-9.4-c1-0-21
Degree $2$
Conductor $5292$
Sign $0.989 + 0.145i$
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 + 2.81·17-s + 4.98·19-s + (0.356 − 0.616i)23-s + (1.05 + 1.82i)25-s + (2.25 + 3.90i)29-s + (2.54 − 4.41i)31-s − 6.87·37-s + (2.93 − 5.08i)41-s + (2.32 + 4.03i)43-s + (−6.49 − 11.2i)47-s − 1.88·53-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 + 0.681·17-s + 1.14·19-s + (0.0742 − 0.128i)23-s + (0.211 + 0.365i)25-s + (0.418 + 0.725i)29-s + (0.457 − 0.793i)31-s − 1.13·37-s + (0.458 − 0.794i)41-s + (0.354 + 0.614i)43-s + (−0.947 − 1.64i)47-s − 0.259·53-s + 0.567·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.989 + 0.145i$
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: \((2,\ 5292,\ (\ :1/2),\ 0.989 + 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.377526610\)
\(L(\frac12)\) \(\approx\) \(2.377526610\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.81T + 17T^{2} \)
19 \( 1 - 4.98T + 19T^{2} \)
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 \( 1 + (-2.54 + 4.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.87T + 37T^{2} \)
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 \( 1 + (6.49 + 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.88T + 53T^{2} \)
59 \( 1 + (7.14 - 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.15 - 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.99 - 6.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + 4.98T + 73T^{2} \)
79 \( 1 + (-4.60 - 7.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.40 + 7.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.65T + 89T^{2} \)
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

−8.252585384227565977845055151572, −7.36475795459891971722764661232, −6.90133086303773131903994077590, −5.81802390744691431726303017334, −5.32831468314596984264467079389, −4.60042915499322147499457265826, −3.69637904613897653384427648252, −2.83294714120292343219820570396, −1.69425205366771891202110922407, −0.909486839005377097679879572640, 0.845552906732558619568844075295, 1.90726765386639271999411019757, 3.08063368537133469302988689958, 3.41150651036523027296848771952, 4.62204412643807633930277450237, 5.33306678020560243500079546322, 6.28219023127570400240663872943, 6.54661381741308516166980364188, 7.55937374873143049175032052258, 8.107376823099475013490852363232

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