Properties

Label 2-30e2-20.3-c1-0-32
Degree $2$
Conductor $900$
Sign $-0.549 + 0.835i$
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  + (−1.19 − 0.760i)2-s + (0.844 + 1.81i)4-s + (0.611 + 0.611i)7-s + (0.371 − 2.80i)8-s − 5.12i·11-s + (−1.76 − 1.76i)13-s + (−0.264 − 1.19i)14-s + (−2.57 + 3.06i)16-s + (−3.76 + 3.76i)17-s + 1.22·19-s + (−3.89 + 6.11i)22-s + (1.07 − 1.07i)23-s + (0.761 + 3.43i)26-s + (−0.592 + 1.62i)28-s − 0.864i·29-s + ⋯
L(s)  = 1  + (−0.843 − 0.537i)2-s + (0.422 + 0.906i)4-s + (0.231 + 0.231i)7-s + (0.131 − 0.991i)8-s − 1.54i·11-s + (−0.488 − 0.488i)13-s + (−0.0706 − 0.319i)14-s + (−0.643 + 0.765i)16-s + (−0.912 + 0.912i)17-s + 0.280·19-s + (−0.831 + 1.30i)22-s + (0.224 − 0.224i)23-s + (0.149 + 0.674i)26-s + (−0.111 + 0.307i)28-s − 0.160i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.549 + 0.835i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.549 + 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.353454 - 0.655275i\)
\(L(\frac12)\) \(\approx\) \(0.353454 - 0.655275i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 + 0.760i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.611 - 0.611i)T + 7iT^{2} \)
11 \( 1 + 5.12iT - 11T^{2} \)
13 \( 1 + (1.76 + 1.76i)T + 13iT^{2} \)
17 \( 1 + (3.76 - 3.76i)T - 17iT^{2} \)
19 \( 1 - 1.22T + 19T^{2} \)
23 \( 1 + (-1.07 + 1.07i)T - 23iT^{2} \)
29 \( 1 + 0.864iT - 29T^{2} \)
31 \( 1 + 7.81iT - 31T^{2} \)
37 \( 1 + (-1.76 + 1.76i)T - 37iT^{2} \)
41 \( 1 + 5.52T + 41T^{2} \)
43 \( 1 + (-6.20 + 6.20i)T - 43iT^{2} \)
47 \( 1 + (2.29 + 2.29i)T + 47iT^{2} \)
53 \( 1 + (2.62 + 2.62i)T + 53iT^{2} \)
59 \( 1 + 0.528T + 59T^{2} \)
61 \( 1 - 4.98T + 61T^{2} \)
67 \( 1 + (6.20 + 6.20i)T + 67iT^{2} \)
71 \( 1 - 8.10iT - 71T^{2} \)
73 \( 1 + (-2.25 - 2.25i)T + 73iT^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 + (-7.95 + 7.95i)T - 83iT^{2} \)
89 \( 1 + 7.25iT - 89T^{2} \)
97 \( 1 + (0.793 - 0.793i)T - 97iT^{2} \)
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

−9.875377439707781126691829823574, −8.858297783803280304010725530567, −8.395308156256568398056289922351, −7.56206660163326632284949107402, −6.47792282172055872542212608430, −5.57660977334623906491623981323, −4.14562710014348324545270558475, −3.13980175067048337126214502736, −2.05834774117868516366409882247, −0.47205819931704010190737449149, 1.48474782716240054844806024777, 2.63279016692604674905004931292, 4.52865388948965708978292104080, 5.06565017605079107437259201796, 6.45320671270648314239082895076, 7.14999033169395120084430458950, 7.68410432593323419278652591373, 8.857044359692314677714524926340, 9.480525986916703007214604412159, 10.14997813198572309123652088210

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