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Properties

Label 2-810-135.122-c1-0-14
Degree $2$
Conductor $810$
Sign $-0.0110 + 0.999i$
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 + (−1.91 + 1.15i)5-s + (−0.957 − 2.05i)7-s + (−0.258 + 0.965i)8-s + (−2.23 − 0.149i)10-s + (−3.39 − 4.04i)11-s + (−2.24 − 3.21i)13-s + (0.393 − 2.23i)14-s + (−0.766 + 0.642i)16-s + (−0.889 − 3.32i)17-s + (3.09 − 1.78i)19-s + (−1.74 − 1.40i)20-s + (−0.459 − 5.25i)22-s + (4.68 + 2.18i)23-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (0.171 + 0.469i)4-s + (−0.855 + 0.517i)5-s + (−0.361 − 0.775i)7-s + (−0.0915 + 0.341i)8-s + (−0.705 − 0.0471i)10-s + (−1.02 − 1.21i)11-s + (−0.623 − 0.890i)13-s + (0.105 − 0.596i)14-s + (−0.191 + 0.160i)16-s + (−0.215 − 0.805i)17-s + (0.709 − 0.409i)19-s + (−0.389 − 0.313i)20-s + (−0.0980 − 1.12i)22-s + (0.976 + 0.455i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0110 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0110 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.0110 + 0.999i$
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: \((2,\ 810,\ (\ :1/2),\ -0.0110 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.616196 - 0.623067i\)
\(L(\frac12)\) \(\approx\) \(0.616196 - 0.623067i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.91 - 1.15i)T \)
good7 \( 1 + (0.957 + 2.05i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (3.39 + 4.04i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (2.24 + 3.21i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (0.889 + 3.32i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.09 + 1.78i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.68 - 2.18i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-0.136 - 0.774i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (10.3 - 3.75i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-8.66 + 2.32i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.74 + 0.484i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.583 - 6.67i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (5.28 - 2.46i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (9.14 + 9.14i)T + 53iT^{2} \)
59 \( 1 + (8.39 + 7.04i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.73 - 0.630i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.63 - 1.84i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (4.51 + 2.60i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.63 + 0.973i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.60 - 1.16i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.91 + 2.72i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (1.05 + 1.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.4 - 1.00i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.22332200201919438793407924494, −9.094394016840301482013164705188, −7.88348586035018376543542475257, −7.50947668976452815786732150441, −6.68315754216282307412682808837, −5.49343389558101877724929949942, −4.72657700698278751779106954812, −3.29631526778871679890171643065, −3.03418430043533466281854084439, −0.34196909115168378712807924107, 1.85541385452201840436787589220, 2.97943996889668623323721169291, 4.23066288102815892937510848517, 4.91061630360637904524631796484, 5.84298923585603550772648215292, 7.09836258488678673135494367681, 7.77123009388454338626072641233, 8.967203352772223501047390014343, 9.587032964631549639478763057971, 10.58563779528373973544974718491

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