↑

Properties

Label 2-648-216.11-c1-0-9
Degree $2$
Conductor $648$
Sign $0.915 - 0.402i$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
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  + (−1.41 − 0.0343i)2-s + (1.99 + 0.0970i)4-s + (2.25 + 0.821i)5-s + (−1.95 − 0.344i)7-s + (−2.82 − 0.205i)8-s + (−3.16 − 1.23i)10-s + (−0.553 − 1.52i)11-s + (3.57 + 4.26i)13-s + (2.75 + 0.554i)14-s + (3.98 + 0.387i)16-s + (6.40 − 3.69i)17-s + (−1.39 + 2.41i)19-s + (4.42 + 1.85i)20-s + (0.730 + 2.16i)22-s + (0.465 + 2.64i)23-s + ⋯
L(s)  = 1  + (−0.999 − 0.0242i)2-s + (0.998 + 0.0485i)4-s + (1.00 + 0.367i)5-s + (−0.738 − 0.130i)7-s + (−0.997 − 0.0727i)8-s + (−0.999 − 0.391i)10-s + (−0.166 − 0.458i)11-s + (0.992 + 1.18i)13-s + (0.735 + 0.148i)14-s + (0.995 + 0.0969i)16-s + (1.55 − 0.897i)17-s + (−0.320 + 0.554i)19-s + (0.989 + 0.415i)20-s + (0.155 + 0.462i)22-s + (0.0970 + 0.550i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.915 - 0.402i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.915 - 0.402i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09733 + 0.230469i\)
\(L(\frac12)\) \(\approx\) \(1.09733 + 0.230469i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0343i)T \)
3 \( 1 \)
good5 \( 1 + (-2.25 - 0.821i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.95 + 0.344i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (0.553 + 1.52i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-3.57 - 4.26i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-6.40 + 3.69i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.39 - 2.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.465 - 2.64i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.138 - 0.116i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.76 + 0.311i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (3.55 - 2.05i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.85 - 3.40i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-11.1 + 4.04i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (2.08 - 11.8i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 + (-1.96 + 5.40i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (-7.60 - 1.34i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.38 + 1.16i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.25 - 2.16i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.79 - 6.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.77 + 8.06i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.718 - 0.855i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (9.87 + 5.70i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.25 - 1.91i)T + (74.3 - 62.3i)T^{2} \)
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

−10.35168856411238188259189874514, −9.699787323489826918554035502516, −9.145292234760941140020107097782, −8.091224273808322650126922557145, −7.06935583931446227843788202444, −6.24237028967913977875646743021, −5.63920903903267543324430309944, −3.66432986028000085933324045081, −2.60852632327071532826354376308, −1.26198504790083149290172262519, 1.01281816541888147832009304398, 2.38629118256085824710387853944, 3.56113431830499327006047075473, 5.53087923757752675570183821618, 5.96269973878373243329523988811, 7.02076761552923016152730986818, 8.109567845349939854746064663754, 8.810051682579710770688379783409, 9.747039595252565479575428507269, 10.24266392993115574437213529734

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