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Properties

Label 2-810-405.23-c1-0-6
Degree $2$
Conductor $810$
Sign $0.368 - 0.929i$
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  + (−0.989 + 0.144i)2-s + (−0.107 − 1.72i)3-s + (0.957 − 0.286i)4-s + (0.876 + 2.05i)5-s + (0.356 + 1.69i)6-s + (−3.77 − 0.109i)7-s + (−0.906 + 0.422i)8-s + (−2.97 + 0.371i)9-s + (−1.16 − 1.90i)10-s + (1.80 − 3.59i)11-s + (−0.598 − 1.62i)12-s + (−1.47 + 0.585i)13-s + (3.75 − 0.438i)14-s + (3.46 − 1.73i)15-s + (0.835 − 0.549i)16-s + (0.414 + 4.73i)17-s + ⋯
L(s)  = 1  + (−0.699 + 0.102i)2-s + (−0.0619 − 0.998i)3-s + (0.478 − 0.143i)4-s + (0.391 + 0.920i)5-s + (0.145 + 0.691i)6-s + (−1.42 − 0.0415i)7-s + (−0.320 + 0.149i)8-s + (−0.992 + 0.123i)9-s + (−0.368 − 0.603i)10-s + (0.544 − 1.08i)11-s + (−0.172 − 0.469i)12-s + (−0.408 + 0.162i)13-s + (1.00 − 0.117i)14-s + (0.893 − 0.448i)15-s + (0.208 − 0.137i)16-s + (0.100 + 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.368 - 0.929i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.368 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557975 + 0.378824i\)
\(L(\frac12)\) \(\approx\) \(0.557975 + 0.378824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.989 - 0.144i)T \)
3 \( 1 + (0.107 + 1.72i)T \)
5 \( 1 + (-0.876 - 2.05i)T \)
good7 \( 1 + (3.77 + 0.109i)T + (6.98 + 0.407i)T^{2} \)
11 \( 1 + (-1.80 + 3.59i)T + (-6.56 - 8.82i)T^{2} \)
13 \( 1 + (1.47 - 0.585i)T + (9.45 - 8.92i)T^{2} \)
17 \( 1 + (-0.414 - 4.73i)T + (-16.7 + 2.95i)T^{2} \)
19 \( 1 + (-4.16 - 4.96i)T + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (0.0834 + 2.86i)T + (-22.9 + 1.33i)T^{2} \)
29 \( 1 + (2.87 + 0.336i)T + (28.2 + 6.68i)T^{2} \)
31 \( 1 + (-6.28 - 1.48i)T + (27.7 + 13.9i)T^{2} \)
37 \( 1 + (-5.55 - 7.93i)T + (-12.6 + 34.7i)T^{2} \)
41 \( 1 + (6.75 - 5.03i)T + (11.7 - 39.2i)T^{2} \)
43 \( 1 + (7.41 - 8.33i)T + (-4.99 - 42.7i)T^{2} \)
47 \( 1 + (-4.19 - 2.58i)T + (21.0 + 42.0i)T^{2} \)
53 \( 1 + (1.58 + 5.90i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.98 + 2.00i)T + (35.2 - 47.3i)T^{2} \)
61 \( 1 + (-1.53 + 5.11i)T + (-50.9 - 33.5i)T^{2} \)
67 \( 1 + (9.46 - 11.9i)T + (-15.4 - 65.1i)T^{2} \)
71 \( 1 + (-4.58 - 12.6i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (-3.78 - 8.11i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (-2.10 - 1.56i)T + (22.6 + 75.6i)T^{2} \)
83 \( 1 + (1.90 + 12.9i)T + (-79.5 + 23.8i)T^{2} \)
89 \( 1 + (-4.50 - 1.64i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-0.122 + 0.591i)T + (-89.0 - 38.4i)T^{2} \)
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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

−10.09461594213131451387019432553, −9.797783114928739643422647773786, −8.559420551779368108094916252972, −7.87956703668907799884834693680, −6.72121623797007372178573699826, −6.40003510218557383237441698867, −5.74158953119992483283732318083, −3.46548516941200369292863503148, −2.79353157679119434949836362997, −1.33145795286591119959804846543, 0.44232413511336212455114586692, 2.42493669905844492496803966356, 3.53747273193298764788535502852, 4.73813837792572532710214326845, 5.56493588818796722375015708573, 6.68921223503037476691431833484, 7.56021294577269823006722018776, 8.965906309647161270898926595352, 9.429762840698098094139217933987, 9.699385227217420714538974538253

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