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Properties

Label 2-2700-45.34-c1-0-8
Degree $2$
Conductor $2700$
Sign $0.972 + 0.232i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)7-s + (1.5 − 2.59i)11-s + (−0.866 + 0.5i)13-s + 6i·17-s + 4·19-s + (2.59 − 1.5i)23-s + (−1.5 + 2.59i)29-s + (−2.5 − 4.33i)31-s − 2i·37-s + (1.5 + 2.59i)41-s + (0.866 + 0.5i)43-s + (7.79 + 4.5i)47-s + (−3 − 5.19i)49-s + 6i·53-s + (1.5 + 2.59i)59-s + ⋯
L(s)  = 1  + (−0.327 − 0.188i)7-s + (0.452 − 0.783i)11-s + (−0.240 + 0.138i)13-s + 1.45i·17-s + 0.917·19-s + (0.541 − 0.312i)23-s + (−0.278 + 0.482i)29-s + (−0.449 − 0.777i)31-s − 0.328i·37-s + (0.234 + 0.405i)41-s + (0.132 + 0.0762i)43-s + (1.13 + 0.656i)47-s + (−0.428 − 0.742i)49-s + 0.824i·53-s + (0.195 + 0.338i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.972 + 0.232i$
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: \((2,\ 2700,\ (\ :1/2),\ 0.972 + 0.232i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.760115242\)
\(L(\frac12)\) \(\approx\) \(1.760115242\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.79 - 4.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.79 + 4.5i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-9.52 - 5.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.893521701767393291862324765596, −8.036854430390009222122484612066, −7.33580517017774476210052546471, −6.41472476230803739521172811380, −5.86411383093590383064641348072, −4.91466259543318138811294963548, −3.86060785697906169053930440476, −3.28155668170697041361954270769, −2.03067928753312742002325838902, −0.798899682111628645875452865526, 0.883927912402901540319852003873, 2.23674300011553588499729938641, 3.12368535933943603941275638930, 4.08586333565605284053376950274, 5.08934890088221056649849747497, 5.60246734878673315859947787575, 6.93509305638667594335096096287, 7.08203668177120011370457903692, 8.068340339683844538063810274338, 9.067359550587830029405615699512

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