↑

Properties

Label 2-30e2-100.83-c1-0-1
Degree $2$
Conductor $900$
Sign $-0.625 - 0.780i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + (−1.26 − 0.642i)2-s + (1.17 + 1.61i)4-s + (0.366 − 2.20i)5-s + (−0.442 − 2.79i)8-s + (−1.87 + 2.54i)10-s + (−5.83 + 2.97i)13-s + (−1.23 + 3.80i)16-s + (−5.34 + 0.846i)17-s + (4.00 − 2.00i)20-s + (−4.73 − 1.61i)25-s + 9.26·26-s + (−0.419 − 0.577i)29-s + (4.00 − 4.00i)32-s + (7.28 + 2.36i)34-s + (3.28 + 6.44i)37-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)2-s + (0.587 + 0.809i)4-s + (0.163 − 0.986i)5-s + (−0.156 − 0.987i)8-s + (−0.593 + 0.804i)10-s + (−1.61 + 0.824i)13-s + (−0.309 + 0.951i)16-s + (−1.29 + 0.205i)17-s + (0.894 − 0.447i)20-s + (−0.946 − 0.323i)25-s + 1.81·26-s + (−0.0779 − 0.107i)29-s + (0.707 − 0.707i)32-s + (1.24 + 0.405i)34-s + (0.539 + 1.05i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.625 - 0.780i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.625 - 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0215201 + 0.0448458i\)
\(L(\frac12)\) \(\approx\) \(0.0215201 + 0.0448458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.642i)T \)
3 \( 1 \)
5 \( 1 + (-0.366 + 2.20i)T \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (5.83 - 2.97i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (5.34 - 0.846i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (0.419 + 0.577i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.28 - 6.44i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (1.04 - 3.22i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (14.2 + 2.26i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.18 - 3.64i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.74 - 13.2i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (17.4 - 5.65i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.07 + 19.4i)T + (-92.2 - 29.9i)T^{2} \)
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

−10.09672999676125365836576154984, −9.591642070821250516767667429744, −8.853139267392375230307901430189, −8.135056815180959411875220812742, −7.18125181784235544892074438864, −6.35168100900149576042142297605, −4.92257555109268031025281714970, −4.17377951877841555068457386755, −2.61623135704315052141536365210, −1.64028179516875072906271738864, 0.02929757167638740175943185869, 2.10545183523479767737322562977, 2.94086267970626867602255614360, 4.65525694853329512346294981567, 5.70629926164404541298198505949, 6.59652176760430387368504234195, 7.34890671595849145085834050297, 7.919078507728821204985781562348, 9.138918326453907804074506933813, 9.716680379779282806148560491710

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