↑

Properties

Label 2-42e2-21.5-c3-0-38
Degree $2$
Conductor $1764$
Sign $-0.974 + 0.224i$
Analytic cond. $104.079$
Root an. cond. $10.2019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (10.7 − 18.6i)5-s + (54.5 − 31.4i)11-s − 28.5i·13-s + (−17.3 − 30.0i)17-s + (−28.9 − 16.7i)19-s + (−104. − 60.1i)23-s + (−168. − 291. i)25-s − 151. i·29-s + (−200. + 115. i)31-s + (−134. + 233. i)37-s + 302.·41-s − 488.·43-s + (−143. + 248. i)47-s + (521. − 300. i)53-s − 1.35e3i·55-s + ⋯
L(s)  = 1  + (0.961 − 1.66i)5-s + (1.49 − 0.863i)11-s − 0.609i·13-s + (−0.247 − 0.428i)17-s + (−0.350 − 0.202i)19-s + (−0.944 − 0.545i)23-s + (−1.34 − 2.33i)25-s − 0.968i·29-s + (−1.15 + 0.669i)31-s + (−0.599 + 1.03i)37-s + 1.15·41-s − 1.73·43-s + (−0.444 + 0.770i)47-s + (1.35 − 0.779i)53-s − 3.31i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.974 + 0.224i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -0.974 + 0.224i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.180408002\)
\(L(\frac12)\) \(\approx\) \(2.180408002\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-10.7 + 18.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-54.5 + 31.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 28.5iT - 2.19e3T^{2} \)
17 \( 1 + (17.3 + 30.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (28.9 + 16.7i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (104. + 60.1i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 151. iT - 2.43e4T^{2} \)
31 \( 1 + (200. - 115. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (134. - 233. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 302.T + 6.89e4T^{2} \)
43 \( 1 + 488.T + 7.95e4T^{2} \)
47 \( 1 + (143. - 248. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-521. + 300. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-9.57 - 16.5i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-421. - 243. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-71.1 - 123. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 281. iT - 3.57e5T^{2} \)
73 \( 1 + (-600. + 346. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-38.1 + 66.0i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 466.T + 5.71e5T^{2} \)
89 \( 1 + (675. - 1.16e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 315. iT - 9.12e5T^{2} \)
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

−8.542052392851950236834325688190, −8.221608580757793218870334436832, −6.74960455664013614604036200025, −6.05166236152214797646006163231, −5.34964031250860720440278109446, −4.51976014825525949674599171891, −3.64770069132737269890986837703, −2.20276459486797420236172048823, −1.25784891195959181770156590312, −0.43753427951405075414026691221, 1.74904448891160885657350260316, 2.10685259845591614013194072826, 3.49780438945025669734688758937, 4.07221409916506405013297680704, 5.50874320952969155782214131103, 6.25871608866132328725622700923, 6.91225512631803134789675106475, 7.33036740795452394548408560170, 8.687962832171431268408231657974, 9.536836225380873840759403187153

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