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Properties

Label 2-810-135.113-c1-0-5
Degree $2$
Conductor $810$
Sign $0.901 - 0.432i$
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 + (−2.23 − 0.0174i)5-s + (1.90 + 1.33i)7-s + (0.965 + 0.258i)8-s + (−2.22 − 0.212i)10-s + (0.889 − 2.44i)11-s + (0.203 + 2.32i)13-s + (1.78 + 1.49i)14-s + (0.939 + 0.342i)16-s + (5.64 − 1.51i)17-s + (2.11 − 1.21i)19-s + (−2.19 − 0.405i)20-s + (1.09 − 2.35i)22-s + (5.43 + 7.76i)23-s + ⋯
L(s)  = 1  + (0.704 + 0.0616i)2-s + (0.492 + 0.0868i)4-s + (−0.999 − 0.00779i)5-s + (0.721 + 0.505i)7-s + (0.341 + 0.0915i)8-s + (−0.703 − 0.0671i)10-s + (0.268 − 0.736i)11-s + (0.0563 + 0.644i)13-s + (0.477 + 0.400i)14-s + (0.234 + 0.0855i)16-s + (1.36 − 0.366i)17-s + (0.484 − 0.279i)19-s + (−0.491 − 0.0906i)20-s + (0.234 − 0.502i)22-s + (1.13 + 1.61i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.901 - 0.432i$
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: \((2,\ 810,\ (\ :1/2),\ 0.901 - 0.432i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23468 + 0.508559i\)
\(L(\frac12)\) \(\approx\) \(2.23468 + 0.508559i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.0174i)T \)
good7 \( 1 + (-1.90 - 1.33i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (-0.889 + 2.44i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.203 - 2.32i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (-5.64 + 1.51i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.11 + 1.21i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.43 - 7.76i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (5.39 - 4.52i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.28 - 7.30i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.48 + 5.55i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.75 + 2.09i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.950 + 2.03i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (-4.66 + 6.65i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (1.68 - 1.68i)T - 53iT^{2} \)
59 \( 1 + (-5.56 + 2.02i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.241 + 1.36i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (11.1 - 0.976i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (-1.67 - 0.967i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.894 - 3.33i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.76 - 3.29i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-1.24 + 14.2i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (4.56 + 7.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.02 + 2.80i)T + (62.3 - 74.3i)T^{2} \)
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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

−10.67609091201265121746959708893, −9.254320832996557970874879326202, −8.589128539891824129436770972293, −7.49736935591677752170055974836, −7.05532090208589947759003089369, −5.53659533942057979451039558315, −5.11626042383857834712223502065, −3.77988326411154995208665174074, −3.14134118498670604046376882257, −1.41129925589652083058999830555, 1.13777781646523427494007266190, 2.82524451328904542039308222666, 3.93377760039424847003995793862, 4.60664838672310905698273818746, 5.59164304669493895196196190099, 6.77827575429567871000415763179, 7.74739736824450854546692476873, 8.030498184343869609622862733438, 9.454072907800900075587358643224, 10.47435425262840834025339384396

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