↑

Properties

Label 2-810-45.34-c1-0-4
Degree $2$
Conductor $810$
Sign $-0.803 - 0.595i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.133 + 2.23i)5-s + (−1.73 − i)7-s + 0.999i·8-s + (−1 + 1.99i)10-s + (−1 + 1.73i)11-s + (−5.19 + 3i)13-s + (−0.999 − 1.73i)14-s + (−0.5 + 0.866i)16-s + 2i·17-s + (−1.86 + 1.23i)20-s + (−1.73 + 0.999i)22-s + (3.46 − 2i)23-s + (−4.96 + 0.598i)25-s − 6·26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.0599 + 0.998i)5-s + (−0.654 − 0.377i)7-s + 0.353i·8-s + (−0.316 + 0.632i)10-s + (−0.301 + 0.522i)11-s + (−1.44 + 0.832i)13-s + (−0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s + 0.485i·17-s + (−0.417 + 0.275i)20-s + (−0.369 + 0.213i)22-s + (0.722 − 0.417i)23-s + (−0.992 + 0.119i)25-s − 1.17·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.803 - 0.595i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ -0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465430 + 1.40877i\)
\(L(\frac12)\) \(\approx\) \(0.465430 + 1.40877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 + (-0.133 - 2.23i)T \)
good7 \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.19 - 3i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-3.46 + 2i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.46 + 2i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.92 - 4i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.92 - 4i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.46 + 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-6.92 - 4i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.40990478634611872529035228044, −10.00784642178723885107752220571, −8.894708552395546871499807248838, −7.63731393262930549307384580684, −6.94242673823540578553136253178, −6.50214875311581585307363303897, −5.22822187667719928742493290796, −4.28469006205999367807306801180, −3.18271539405182893107735811771, −2.24935550316776924702207370883, 0.56578773997756096362111234065, 2.34799070293649203093814176380, 3.29609372545562301919233635451, 4.62670169705209815385098383564, 5.31998575769499049995256797310, 6.09245236927974655739083009946, 7.33666129176799793675013286716, 8.216778683629635457076122025156, 9.363678836253141974829863914034, 9.770632756602790027042918334187

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