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Properties

Label 2-810-15.14-c2-0-40
Degree $2$
Conductor $810$
Sign $-0.0840 + 0.996i$
Analytic cond. $22.0709$
Root an. cond. $4.69796$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s + (0.420 − 4.98i)5-s + 1.96i·7-s + 2.82·8-s + (0.594 − 7.04i)10-s − 3.81i·11-s − 18.6i·13-s + 2.77i·14-s + 4.00·16-s + 17.6·17-s − 26.0·19-s + (0.840 − 9.96i)20-s − 5.39i·22-s − 9.64·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.500·4-s + (0.0840 − 0.996i)5-s + 0.280i·7-s + 0.353·8-s + (0.0594 − 0.704i)10-s − 0.346i·11-s − 1.43i·13-s + 0.198i·14-s + 0.250·16-s + 1.03·17-s − 1.37·19-s + (0.0420 − 0.498i)20-s − 0.245i·22-s − 0.419·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0840 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0840 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.0840 + 0.996i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ -0.0840 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.588588153\)
\(L(\frac12)\) \(\approx\) \(2.588588153\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 + (-0.420 + 4.98i)T \)
good7 \( 1 - 1.96iT - 49T^{2} \)
11 \( 1 + 3.81iT - 121T^{2} \)
13 \( 1 + 18.6iT - 169T^{2} \)
17 \( 1 - 17.6T + 289T^{2} \)
19 \( 1 + 26.0T + 361T^{2} \)
23 \( 1 + 9.64T + 529T^{2} \)
29 \( 1 - 6.01iT - 841T^{2} \)
31 \( 1 - 25.3T + 961T^{2} \)
37 \( 1 + 39.9iT - 1.36e3T^{2} \)
41 \( 1 + 35.4iT - 1.68e3T^{2} \)
43 \( 1 + 61.0iT - 1.84e3T^{2} \)
47 \( 1 + 70.8T + 2.20e3T^{2} \)
53 \( 1 - 83.3T + 2.80e3T^{2} \)
59 \( 1 - 37.3iT - 3.48e3T^{2} \)
61 \( 1 + 4.75T + 3.72e3T^{2} \)
67 \( 1 - 9.27iT - 4.48e3T^{2} \)
71 \( 1 - 34.1iT - 5.04e3T^{2} \)
73 \( 1 + 22.9iT - 5.32e3T^{2} \)
79 \( 1 + 99.0T + 6.24e3T^{2} \)
83 \( 1 + 37.9T + 6.88e3T^{2} \)
89 \( 1 + 43.0iT - 7.92e3T^{2} \)
97 \( 1 + 27.4iT - 9.40e3T^{2} \)
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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

−10.03710804595791351213416232752, −8.753116741180331739476378962460, −8.231265625342991055641566114994, −7.23167333000987328896735852668, −5.83039774531875580105014191168, −5.55146802340971724402530395935, −4.42336739047235641284628483319, −3.43278395535829973389453980200, −2.16523598242453692375203079325, −0.67076553226849649049683241826, 1.73444500976964783010624195712, 2.82975403635055171753155502717, 3.94008926137861782124787637770, 4.71371254585893352689399549297, 6.12302462718022931716402544972, 6.60130965383397755628556354942, 7.47084170167680197733685914766, 8.423074470702124589239106611266, 9.786494360687747809999610895017, 10.23690924629325638554795084817

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