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Properties

Label 2-5292-9.4-c1-0-23
Degree $2$
Conductor $5292$
Sign $0.928 + 0.371i$
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  + (1.94 − 3.37i)5-s + (2.18 + 3.78i)11-s + (−0.792 + 1.37i)13-s + 5.33·17-s − 4.64·19-s + (0.183 − 0.318i)23-s + (−5.07 − 8.79i)25-s + (5.08 + 8.81i)29-s + (1.14 − 1.98i)31-s + 10.8·37-s + (0.690 − 1.19i)41-s + (3.81 + 6.61i)43-s + (−3.80 − 6.58i)47-s + 0.925·53-s + 17.0·55-s + ⋯
L(s)  = 1  + (0.870 − 1.50i)5-s + (0.659 + 1.14i)11-s + (−0.219 + 0.380i)13-s + 1.29·17-s − 1.06·19-s + (0.0383 − 0.0664i)23-s + (−1.01 − 1.75i)25-s + (0.944 + 1.63i)29-s + (0.206 − 0.357i)31-s + 1.78·37-s + (0.107 − 0.186i)41-s + (0.582 + 1.00i)43-s + (−0.554 − 0.961i)47-s + 0.127·53-s + 2.29·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.928 + 0.371i$
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: \((2,\ 5292,\ (\ :1/2),\ 0.928 + 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.589920816\)
\(L(\frac12)\) \(\approx\) \(2.589920816\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 + 3.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.792 - 1.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.33T + 17T^{2} \)
19 \( 1 + 4.64T + 19T^{2} \)
23 \( 1 + (-0.183 + 0.318i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.08 - 8.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.14 + 1.98i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 + (-0.690 + 1.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.81 - 6.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.80 + 6.58i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.925T + 53T^{2} \)
59 \( 1 + (0.460 - 0.797i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.27 + 5.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.50 - 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.91T + 71T^{2} \)
73 \( 1 - 7.56T + 73T^{2} \)
79 \( 1 + (0.987 + 1.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.253 - 0.438i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + (-4.45 - 7.71i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.266897219230684081090315462929, −7.51467560706438317602817537490, −6.57086193760047084163211043304, −5.99745402260862591804948063610, −5.06048257605267736581024313889, −4.65906679062242654198244971451, −3.89405363794656840669277229520, −2.55114032450808069368796815980, −1.65876900884437433584702086088, −0.967597710416652193236394051550, 0.864595929329807412601138768252, 2.15560738196045678047673368989, 2.90072491885599893619297567392, 3.50427417011230386875216486448, 4.49962889139576392205181393986, 5.71527150977619267520991763859, 6.14761854583483919240322108383, 6.54148589494891498467609808109, 7.57308337770889711317870029681, 8.063740219149057541313030307357

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