↑

Properties

Label 2-810-135.122-c1-0-7
Degree $2$
Conductor $810$
Sign $0.688 + 0.725i$
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  + (−0.819 − 0.573i)2-s + (0.342 + 0.939i)4-s + (−2.07 + 0.845i)5-s + (0.101 + 0.217i)7-s + (0.258 − 0.965i)8-s + (2.18 + 0.494i)10-s + (−2.07 − 2.47i)11-s + (1.10 + 1.57i)13-s + (0.0417 − 0.236i)14-s + (−0.766 + 0.642i)16-s + (−0.935 − 3.48i)17-s + (1.18 − 0.683i)19-s + (−1.50 − 1.65i)20-s + (0.281 + 3.21i)22-s + (6.16 + 2.87i)23-s + ⋯
L(s)  = 1  + (−0.579 − 0.405i)2-s + (0.171 + 0.469i)4-s + (−0.925 + 0.378i)5-s + (0.0383 + 0.0823i)7-s + (0.0915 − 0.341i)8-s + (0.689 + 0.156i)10-s + (−0.625 − 0.745i)11-s + (0.306 + 0.437i)13-s + (0.0111 − 0.0632i)14-s + (−0.191 + 0.160i)16-s + (−0.226 − 0.846i)17-s + (0.271 − 0.156i)19-s + (−0.335 − 0.370i)20-s + (0.0599 + 0.685i)22-s + (1.28 + 0.599i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.688 + 0.725i$
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: \((2,\ 810,\ (\ :1/2),\ 0.688 + 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.813909 - 0.349559i\)
\(L(\frac12)\) \(\approx\) \(0.813909 - 0.349559i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 - 0.845i)T \)
good7 \( 1 + (-0.101 - 0.217i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (2.07 + 2.47i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-1.10 - 1.57i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (0.935 + 3.48i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.18 + 0.683i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.16 - 2.87i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-0.340 - 1.92i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.42 + 0.881i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-5.53 + 1.48i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.258 + 0.0455i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.573 + 6.54i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-11.9 + 5.56i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-9.15 - 9.15i)T + 53iT^{2} \)
59 \( 1 + (-4.98 - 4.18i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (14.0 + 5.13i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (3.34 - 2.34i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (9.56 + 5.52i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.32 - 0.623i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.06 + 0.364i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-5.39 + 7.69i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (1.90 + 3.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.17 + 0.627i)T + (95.5 + 16.8i)T^{2} \)
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

−10.35065191535264692936918813204, −9.085666651442330158011082502275, −8.656224425324581534876195622144, −7.50614573113008858025653856687, −7.10054188582902241828664664199, −5.77013862030365873024988046438, −4.55021063034157453616524033609, −3.42783869526529876372874195802, −2.57901125544809184528306762422, −0.71387179873845601033383829080, 1.00692079396925338513844010052, 2.72203823097237944862842218439, 4.14248419385253631166571822059, 4.99698696693952518817940812754, 6.10182304256954786874073946650, 7.18336142390425788410786620697, 7.83356266921364276165205641567, 8.536579855132535142913624700872, 9.353818629906434658023514366708, 10.42121642388789185954677875160

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