↑

Properties

Label 2-5292-63.16-c1-0-24
Degree $2$
Conductor $5292$
Sign $0.00214 + 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.38·5-s + 2.25·11-s + (−2.37 − 4.12i)13-s + (2.15 + 3.72i)17-s + (−4.29 + 7.44i)19-s − 1.32·23-s + 0.687·25-s + (3.87 − 6.71i)29-s + (0.405 − 0.702i)31-s + (2.31 − 4.01i)37-s + (5.00 + 8.66i)41-s + (−1.74 + 3.01i)43-s + (2.18 + 3.78i)47-s + (−5.83 − 10.1i)53-s − 5.38·55-s + ⋯
L(s)  = 1  − 1.06·5-s + 0.681·11-s + (−0.659 − 1.14i)13-s + (0.521 + 0.904i)17-s + (−0.986 + 1.70i)19-s − 0.277·23-s + 0.137·25-s + (0.720 − 1.24i)29-s + (0.0727 − 0.126i)31-s + (0.380 − 0.659i)37-s + (0.781 + 1.35i)41-s + (−0.265 + 0.460i)43-s + (0.318 + 0.551i)47-s + (−0.802 − 1.38i)53-s − 0.726·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00214 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00214 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.00214 + 0.999i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (3313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.00214 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8766507328\)
\(L(\frac12)\) \(\approx\) \(0.8766507328\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.38T + 5T^{2} \)
11 \( 1 - 2.25T + 11T^{2} \)
13 \( 1 + (2.37 + 4.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.15 - 3.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.29 - 7.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.32T + 23T^{2} \)
29 \( 1 + (-3.87 + 6.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.405 + 0.702i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.31 + 4.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.00 - 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.74 - 3.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.18 - 3.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.83 + 10.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.40 + 4.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.575 + 0.997i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.06 + 3.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.41T + 71T^{2} \)
73 \( 1 + (6.05 + 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.23 - 7.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.817 + 1.41i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.17 + 5.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.98 - 10.3i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.938096567177069087881309093441, −7.69199021975339271794065220953, −6.37812794768270461896674538118, −6.07166910632970614336107580618, −5.02102906708690991320198357547, −4.05924187272622083292698368792, −3.74695733491934675087217306250, −2.70430488087844083841644677620, −1.55422627634711711586510543043, −0.29505810569427174215793276049, 0.895876447417856461137589422179, 2.23180338567001611009296520295, 3.07729728563457624391071591162, 4.10714359535305166004558348868, 4.51229098428811276512081141522, 5.31615065606725474155468874671, 6.49169121186504559615800334297, 7.03350358637818237008873792901, 7.46252576736830920466793532559, 8.468819764584945182671394499148

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