↑

Properties

Label 2-30e2-9.2-c2-0-29
Degree $2$
Conductor $900$
Sign $0.213 + 0.976i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.69 − 1.32i)3-s + (−1.36 + 2.35i)7-s + (5.49 − 7.12i)9-s + (−2.77 − 1.60i)11-s + (−3.99 − 6.92i)13-s − 1.82i·17-s + 15.4·19-s + (−0.544 + 8.14i)21-s + (28.0 − 16.1i)23-s + (5.37 − 26.4i)27-s + (−25.5 − 14.7i)29-s + (−26.5 − 45.9i)31-s + (−9.58 − 0.641i)33-s + 30.0·37-s + (−19.9 − 13.3i)39-s + ⋯
L(s)  = 1  + (0.897 − 0.441i)3-s + (−0.194 + 0.336i)7-s + (0.610 − 0.791i)9-s + (−0.252 − 0.145i)11-s + (−0.307 − 0.532i)13-s − 0.107i·17-s + 0.812·19-s + (−0.0259 + 0.387i)21-s + (1.21 − 0.703i)23-s + (0.199 − 0.979i)27-s + (−0.881 − 0.509i)29-s + (−0.854 − 1.48i)31-s + (−0.290 − 0.0194i)33-s + 0.812·37-s + (−0.510 − 0.342i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.213 + 0.976i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.213 + 0.976i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.390338269\)
\(L(\frac12)\) \(\approx\) \(2.390338269\)
\(L(2)\) 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 \( 1 + (-2.69 + 1.32i)T \)
5 \( 1 \)
good7 \( 1 + (1.36 - 2.35i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (2.77 + 1.60i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.99 + 6.92i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 1.82iT - 289T^{2} \)
19 \( 1 - 15.4T + 361T^{2} \)
23 \( 1 + (-28.0 + 16.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (25.5 + 14.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (26.5 + 45.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 30.0T + 1.36e3T^{2} \)
41 \( 1 + (-66.5 + 38.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (4.49 - 7.77i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (10.2 + 5.92i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 28.9iT - 2.80e3T^{2} \)
59 \( 1 + (16.6 - 9.60i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-30.9 + 53.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.59 + 4.49i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 96.0iT - 5.04e3T^{2} \)
73 \( 1 + 127.T + 5.32e3T^{2} \)
79 \( 1 + (-24.2 + 41.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-83.1 - 48.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 21.4iT - 7.92e3T^{2} \)
97 \( 1 + (83.3 - 144. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−9.330826039552187921168313396078, −9.145174042408081312183077373369, −7.80682894835214152011547577321, −7.53205844390448507121153414852, −6.34378257193730673085786819435, −5.42798016434811353845551522516, −4.16223547669643402837592013975, −3.05962194098420731589816899893, −2.27092309965738252533090357417, −0.72348077262194157035522845172, 1.44435981348617919399426321331, 2.78239163385470486099657379678, 3.64321349262694554479350033115, 4.66458294518270947119745751662, 5.55085084941615719081752443057, 7.07473470749067185707417095934, 7.43618242861063775097457758645, 8.572825928303804580958059909779, 9.303753514527821419811367013199, 9.892842828733058217517279115632

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