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Properties

Label 2-810-135.122-c1-0-15
Degree $2$
Conductor $810$
Sign $0.543 + 0.839i$
Analytic cond. $6.46788$
Root an. cond. $2.54320$
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  + (0.819 + 0.573i)2-s + (0.342 + 0.939i)4-s + (1.52 − 1.63i)5-s + (−1.90 − 4.08i)7-s + (−0.258 + 0.965i)8-s + (2.18 − 0.470i)10-s + (−2.45 − 2.92i)11-s + (2.63 + 3.76i)13-s + (0.783 − 4.44i)14-s + (−0.766 + 0.642i)16-s + (−0.996 − 3.72i)17-s + (1.79 − 1.03i)19-s + (2.06 + 0.868i)20-s + (−0.332 − 3.79i)22-s + (−5.25 − 2.45i)23-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (0.171 + 0.469i)4-s + (0.680 − 0.733i)5-s + (−0.720 − 1.54i)7-s + (−0.0915 + 0.341i)8-s + (0.691 − 0.148i)10-s + (−0.739 − 0.880i)11-s + (0.730 + 1.04i)13-s + (0.209 − 1.18i)14-s + (−0.191 + 0.160i)16-s + (−0.241 − 0.902i)17-s + (0.411 − 0.237i)19-s + (0.460 + 0.194i)20-s + (−0.0708 − 0.810i)22-s + (−1.09 − 0.511i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.543 + 0.839i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75264 - 0.953362i\)
\(L(\frac12)\) \(\approx\) \(1.75264 - 0.953362i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.819 - 0.573i)T \)
3 \( 1 \)
5 \( 1 + (-1.52 + 1.63i)T \)
good7 \( 1 + (1.90 + 4.08i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (2.45 + 2.92i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-2.63 - 3.76i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (0.996 + 3.72i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.79 + 1.03i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.25 + 2.45i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.501 + 2.84i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.26 + 0.825i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-5.03 + 1.34i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.59 - 1.34i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.879 - 10.0i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-2.65 + 1.23i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-0.405 - 0.405i)T + 53iT^{2} \)
59 \( 1 + (-6.05 - 5.07i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-12.9 - 4.72i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (3.68 - 2.58i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (9.41 + 5.43i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.262 + 0.0703i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-10.2 + 1.80i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.680 - 0.972i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (-2.08 - 3.60i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.69 + 0.673i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.03833662474837876236044008613, −9.318689561052974771584921703713, −8.302619072738161228110603600895, −7.42607089095829105254485254631, −6.45038553289828399544221934699, −5.84473964529030171339050511039, −4.59846866296611275507436929242, −3.94879545748573099990703140099, −2.63343667807571086781417567272, −0.814180516316998554595369790641, 2.00105282755089765306529173866, 2.75664803447997995102534915004, 3.73069393148123279161345714302, 5.37402644846531945071442863913, 5.78557833335959311710297302162, 6.57518501416653084879904968044, 7.80375641589105935798778121132, 8.864928201024762520695689186349, 9.850855529481694556970419546774, 10.28061164962903792012921140954

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