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Properties

Label 2-810-135.122-c1-0-13
Degree $2$
Conductor $810$
Sign $-0.951 + 0.306i$
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 + (−2.18 + 0.473i)5-s + (−0.171 − 0.368i)7-s + (0.258 − 0.965i)8-s + (2.06 + 0.865i)10-s + (3.16 + 3.76i)11-s + (−2.75 − 3.93i)13-s + (−0.0706 + 0.400i)14-s + (−0.766 + 0.642i)16-s + (−0.352 − 1.31i)17-s + (−0.763 + 0.440i)19-s + (−1.19 − 1.89i)20-s + (−0.428 − 4.89i)22-s + (−0.893 − 0.416i)23-s + ⋯
L(s)  = 1  + (−0.579 − 0.405i)2-s + (0.171 + 0.469i)4-s + (−0.977 + 0.211i)5-s + (−0.0649 − 0.139i)7-s + (0.0915 − 0.341i)8-s + (0.652 + 0.273i)10-s + (0.952 + 1.13i)11-s + (−0.764 − 1.09i)13-s + (−0.0188 + 0.107i)14-s + (−0.191 + 0.160i)16-s + (−0.0856 − 0.319i)17-s + (−0.175 + 0.101i)19-s + (−0.266 − 0.422i)20-s + (−0.0913 − 1.04i)22-s + (−0.186 − 0.0868i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.951 + 0.306i$
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: \((2,\ 810,\ (\ :1/2),\ -0.951 + 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0427898 - 0.272582i\)
\(L(\frac12)\) \(\approx\) \(0.0427898 - 0.272582i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 - 0.473i)T \)
good7 \( 1 + (0.171 + 0.368i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (-3.16 - 3.76i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (2.75 + 3.93i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (0.352 + 1.31i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.763 - 0.440i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.893 + 0.416i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (1.09 + 6.18i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (7.20 - 2.62i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (8.35 - 2.23i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (10.9 + 1.92i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.07 + 12.3i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (3.46 - 1.61i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (0.890 + 0.890i)T + 53iT^{2} \)
59 \( 1 + (0.0286 + 0.0240i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-3.45 - 1.25i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.00 - 1.40i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (4.92 + 2.84i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.355 + 0.0951i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.82 - 0.320i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.186 + 0.266i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (6.26 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.287 + 0.0251i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.04834972621507864936998917633, −9.029349947636566198802999478657, −8.214895041417763993047442772778, −7.25889678275057647581291436655, −6.88933843939986826102277985900, −5.27163696110789394880568999302, −4.15497834711261664620313873414, −3.30678445350844577839213252678, −1.95273504039734495543673228931, −0.16670012091946429622068219933, 1.55386026597330761504303514915, 3.30884599653132839680784367704, 4.30365865002348835532853168160, 5.41750413877388640702474196438, 6.57171255077768527869787797276, 7.17413261014069161216535195980, 8.208450892301406834801406948588, 8.886107926176318071215972465790, 9.461845422739536763766452744076, 10.70765363567702892862854548682

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