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Properties

Label 2-810-135.122-c1-0-4
Degree $2$
Conductor $810$
Sign $-0.589 - 0.807i$
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 + (−0.160 + 2.23i)5-s + (1.92 + 4.12i)7-s + (−0.258 + 0.965i)8-s + (−1.41 + 1.73i)10-s + (0.0108 + 0.0129i)11-s + (−1.70 − 2.43i)13-s + (−0.790 + 4.48i)14-s + (−0.766 + 0.642i)16-s + (−0.126 − 0.470i)17-s + (2.44 − 1.40i)19-s + (−2.15 + 0.612i)20-s + (0.00146 + 0.0167i)22-s + (−5.98 − 2.79i)23-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (0.171 + 0.469i)4-s + (−0.0716 + 0.997i)5-s + (0.726 + 1.55i)7-s + (−0.0915 + 0.341i)8-s + (−0.446 + 0.548i)10-s + (0.00326 + 0.00389i)11-s + (−0.472 − 0.674i)13-s + (−0.211 + 1.19i)14-s + (−0.191 + 0.160i)16-s + (−0.0305 − 0.114i)17-s + (0.559 − 0.323i)19-s + (−0.480 + 0.136i)20-s + (0.000313 + 0.00358i)22-s + (−1.24 − 0.582i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.589 - 0.807i$
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: \((2,\ 810,\ (\ :1/2),\ -0.589 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.940519 + 1.85046i\)
\(L(\frac12)\) \(\approx\) \(0.940519 + 1.85046i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.160 - 2.23i)T \)
good7 \( 1 + (-1.92 - 4.12i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (-0.0108 - 0.0129i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (1.70 + 2.43i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (0.126 + 0.470i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.44 + 1.40i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.98 + 2.79i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-1.01 - 5.76i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (4.13 - 1.50i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-9.53 + 2.55i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-9.02 - 1.59i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.278 + 3.18i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-2.19 + 1.02i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-2.06 - 2.06i)T + 53iT^{2} \)
59 \( 1 + (-10.6 - 8.91i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (5.02 + 1.82i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (8.68 - 6.07i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-12.0 - 6.92i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.59 - 0.695i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.13 + 0.905i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.30 + 3.28i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (8.88 + 15.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.2 + 1.25i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.73914782511408927461742512438, −9.659979870576012110355522045746, −8.684447620545161350953651204362, −7.84893692689977316299678395007, −7.09945684709303767202866652050, −5.92533780581365537676553586847, −5.48558264000739777970086027463, −4.30849538410019220963965933498, −2.95318015838729796976706352638, −2.26608017305029311402379706441, 0.876748287995542307898860680112, 2.02917277031408292127348550391, 3.89300476482247362519941815197, 4.30071656120632535000106108505, 5.23391657536207999815613002172, 6.30927337322297508796477652739, 7.59948121954986880231088103222, 7.971923683183069353688955022028, 9.424481023351590456436832920544, 9.910423451166472699571298237667

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