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Properties

Label 2-810-135.113-c1-0-10
Degree $2$
Conductor $810$
Sign $-0.168 + 0.985i$
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  + (−0.996 − 0.0871i)2-s + (0.984 + 0.173i)4-s + (−2.19 + 0.428i)5-s + (0.926 + 0.648i)7-s + (−0.965 − 0.258i)8-s + (2.22 − 0.235i)10-s + (1.09 − 2.99i)11-s + (−0.104 − 1.18i)13-s + (−0.866 − 0.726i)14-s + (0.939 + 0.342i)16-s + (−3.07 + 0.824i)17-s + (−4.45 + 2.57i)19-s + (−2.23 + 0.0412i)20-s + (−1.34 + 2.89i)22-s + (0.618 + 0.882i)23-s + ⋯
L(s)  = 1  + (−0.704 − 0.0616i)2-s + (0.492 + 0.0868i)4-s + (−0.981 + 0.191i)5-s + (0.350 + 0.245i)7-s + (−0.341 − 0.0915i)8-s + (0.703 − 0.0745i)10-s + (0.329 − 0.904i)11-s + (−0.0288 − 0.330i)13-s + (−0.231 − 0.194i)14-s + (0.234 + 0.0855i)16-s + (−0.746 + 0.199i)17-s + (−1.02 + 0.590i)19-s + (−0.499 + 0.00921i)20-s + (−0.287 + 0.616i)22-s + (0.128 + 0.184i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.168 + 0.985i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ -0.168 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.386774 - 0.458681i\)
\(L(\frac12)\) \(\approx\) \(0.386774 - 0.458681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.996 + 0.0871i)T \)
3 \( 1 \)
5 \( 1 + (2.19 - 0.428i)T \)
good7 \( 1 + (-0.926 - 0.648i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (-1.09 + 2.99i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.104 + 1.18i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (3.07 - 0.824i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.45 - 2.57i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.618 - 0.882i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (-4.18 + 3.50i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.18 + 6.70i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.00 + 7.48i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.43 + 4.09i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.90 + 6.23i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (2.77 - 3.96i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (3.59 - 3.59i)T - 53iT^{2} \)
59 \( 1 + (-5.86 + 2.13i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (2.36 + 13.3i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-4.47 + 0.391i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (5.95 + 3.43i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.81 - 10.4i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.98 - 2.36i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-1.48 + 17.0i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (7.57 + 13.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.363 - 0.169i)T + (62.3 - 74.3i)T^{2} \)
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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

−10.09266040857501588558717875931, −8.915591359496833193659544758738, −8.372225004025927492415023702990, −7.69749349164369202711790381731, −6.66525837402277738839793067363, −5.81546772243645849880358409366, −4.39644231005065072369143916485, −3.46021357129906727511650502127, −2.16199158933845267694402673289, −0.39598853936804147630576109098, 1.37395457184876111106701489678, 2.84450315362965224574263004062, 4.29523923792330904993685290015, 4.87901695931561772195404579047, 6.66435056967810134192593364955, 6.97460299110538881301093213840, 8.123132260369959996652963953834, 8.650445385025865937389835359211, 9.535502044140344228097440896294, 10.54553005002316453017239232429

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