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Properties

Label 2-810-135.122-c1-0-10
Degree $2$
Conductor $810$
Sign $0.595 - 0.803i$
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 + (1.67 − 1.47i)5-s + (1.22 + 2.62i)7-s + (−0.258 + 0.965i)8-s + (2.22 − 0.247i)10-s + (4.04 + 4.81i)11-s + (−2.05 − 2.93i)13-s + (−0.503 + 2.85i)14-s + (−0.766 + 0.642i)16-s + (−1.42 − 5.31i)17-s + (−3.76 + 2.17i)19-s + (1.96 + 1.07i)20-s + (0.548 + 6.26i)22-s + (4.79 + 2.23i)23-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (0.171 + 0.469i)4-s + (0.750 − 0.660i)5-s + (0.462 + 0.992i)7-s + (−0.0915 + 0.341i)8-s + (0.702 − 0.0783i)10-s + (1.21 + 1.45i)11-s + (−0.570 − 0.815i)13-s + (−0.134 + 0.762i)14-s + (−0.191 + 0.160i)16-s + (−0.345 − 1.28i)17-s + (−0.864 + 0.499i)19-s + (0.438 + 0.239i)20-s + (0.116 + 1.33i)22-s + (0.998 + 0.465i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.595 - 0.803i$
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: \((2,\ 810,\ (\ :1/2),\ 0.595 - 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30508 + 1.16058i\)
\(L(\frac12)\) \(\approx\) \(2.30508 + 1.16058i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 + 1.47i)T \)
good7 \( 1 + (-1.22 - 2.62i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (-4.04 - 4.81i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (2.05 + 2.93i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (1.42 + 5.31i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.76 - 2.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.79 - 2.23i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-0.590 - 3.34i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.02 - 0.736i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-1.56 + 0.419i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.83 - 0.500i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.447 - 5.11i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-8.80 + 4.10i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-3.10 - 3.10i)T + 53iT^{2} \)
59 \( 1 + (6.77 + 5.68i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (3.18 + 1.15i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (4.27 - 2.99i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (14.1 + 8.15i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.13 - 1.37i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.25 - 0.926i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-7.05 + 10.0i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (4.60 + 7.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.93 + 0.344i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.25044906232591337831724408117, −9.217725390248776833848111808489, −8.929973227241039475291129481376, −7.65812852926435476151064847816, −6.83181184065801662843604916797, −5.81830646221862960934503438917, −5.02521825553429461319194546247, −4.40648451068133834064158208735, −2.74544510657277963637474044504, −1.70494511372736557856733593958, 1.25224437671798320218999583575, 2.49165192190692712381763217721, 3.78042534166526047614287044018, 4.43917397527242353353867949287, 5.84982245219126131692735313930, 6.48741923436705905965359465303, 7.24039149310222496134453165053, 8.648328265465487878494853538431, 9.331223150054955481717266365827, 10.53955662877134124834182215465

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