↑

Properties

Label 2-2700-45.34-c1-0-16
Degree $2$
Conductor $2700$
Sign $-0.861 + 0.506i$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
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  + (−2.36 − 1.36i)7-s + (2.76 − 4.78i)11-s + (3.05 − 1.76i)13-s + 5.52i·17-s − 7.52·19-s + (0.635 − 0.367i)23-s + (2.23 − 3.86i)29-s + (−3.76 − 6.51i)31-s + 6.05i·37-s + (−0.527 − 0.914i)41-s + (−3.05 − 1.76i)43-s + (1.04 + 0.604i)47-s + (0.237 + 0.412i)49-s + 1.46i·53-s + (0.734 + 1.27i)59-s + ⋯
L(s)  = 1  + (−0.894 − 0.516i)7-s + (0.832 − 1.44i)11-s + (0.846 − 0.488i)13-s + 1.33i·17-s − 1.72·19-s + (0.132 − 0.0765i)23-s + (0.414 − 0.718i)29-s + (−0.675 − 1.17i)31-s + 0.995i·37-s + (−0.0824 − 0.142i)41-s + (−0.465 − 0.268i)43-s + (0.152 + 0.0882i)47-s + (0.0339 + 0.0588i)49-s + 0.201i·53-s + (0.0955 + 0.165i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.861 + 0.506i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ -0.861 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8028665690\)
\(L(\frac12)\) \(\approx\) \(0.8028665690\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (2.36 + 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.76 + 4.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.05 + 1.76i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.52iT - 17T^{2} \)
19 \( 1 + 7.52T + 19T^{2} \)
23 \( 1 + (-0.635 + 0.367i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.23 + 3.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.76 + 6.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.05iT - 37T^{2} \)
41 \( 1 + (0.527 + 0.914i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.05 + 1.76i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.04 - 0.604i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.46iT - 53T^{2} \)
59 \( 1 + (-0.734 - 1.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.52 - 7.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.68 - 2.12i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.56 - 2.63i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (8.19 + 4.73i)T + (48.5 + 84.0i)T^{2} \)
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

−8.534864769481352679743361342799, −7.975460953370304632971460405567, −6.74167907154912982742537766164, −6.18816575873329404489724878183, −5.80150092578306492648319840408, −4.21425764398865619060022694080, −3.78807116425407789066503341089, −2.91322694685320716698636122301, −1.49637672813476961713601589032, −0.25966647738693732126749271007, 1.53517632744931112173531969545, 2.51228158634158376264165399294, 3.56808730837991258835558516783, 4.40439213517879183285587033038, 5.21142217042885327823939538497, 6.40034516886784742036592330087, 6.68665668901677733292038890768, 7.47043830655675688927597079884, 8.703516282387343834488934857913, 9.100431218763394875954830132199

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