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Properties

Label 2-42e2-63.5-c1-0-20
Degree $2$
Conductor $1764$
Sign $0.910 - 0.414i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (1.70 + 0.287i)3-s + (−0.266 − 0.462i)5-s + (2.83 + 0.983i)9-s + (3.39 + 1.96i)11-s + (0.116 + 0.0674i)13-s + (−0.322 − 0.866i)15-s + (2.16 + 3.74i)17-s + (−1.93 − 1.11i)19-s + (1.70 − 0.983i)23-s + (2.35 − 4.08i)25-s + (4.55 + 2.49i)27-s + (−5.16 + 2.98i)29-s − 0.924i·31-s + (5.24 + 4.33i)33-s + (−3.89 + 6.75i)37-s + ⋯
L(s)  = 1  + (0.986 + 0.166i)3-s + (−0.119 − 0.206i)5-s + (0.944 + 0.327i)9-s + (1.02 + 0.591i)11-s + (0.0324 + 0.0187i)13-s + (−0.0832 − 0.223i)15-s + (0.524 + 0.908i)17-s + (−0.442 − 0.255i)19-s + (0.355 − 0.205i)23-s + (0.471 − 0.816i)25-s + (0.877 + 0.480i)27-s + (−0.959 + 0.553i)29-s − 0.165i·31-s + (0.912 + 0.753i)33-s + (−0.641 + 1.11i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.910 - 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.676191678\)
\(L(\frac12)\) \(\approx\) \(2.676191678\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.287i)T \)
7 \( 1 \)
good5 \( 1 + (0.266 + 0.462i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.39 - 1.96i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.116 - 0.0674i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.16 - 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.93 + 1.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.70 + 0.983i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.16 - 2.98i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.924iT - 31T^{2} \)
37 \( 1 + (3.89 - 6.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.59 + 7.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.24 - 5.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.08T + 47T^{2} \)
53 \( 1 + (-9.54 + 5.50i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.79T + 59T^{2} \)
61 \( 1 - 10.7iT - 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 3.22iT - 71T^{2} \)
73 \( 1 + (-0.329 + 0.190i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.20T + 79T^{2} \)
83 \( 1 + (-1.28 - 2.21i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.56 + 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.6 - 7.89i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.162524511803248956394440873945, −8.689403919764235077591345134642, −7.898704115449006870741999279389, −7.05681776261119735431811624244, −6.33460274131644581309108528526, −5.07063589318213869274571189687, −4.18073751751971374021115456004, −3.54854563421827939298584991514, −2.34631145853591824246889842973, −1.32635802039883100932476658681, 1.06210670258376150423727226397, 2.28055663135964647972002781660, 3.37561886454766920970662871877, 3.90946067959595679275746550310, 5.11174143662097114620819398708, 6.18920620859077580775933085673, 7.04781346146312489519113091224, 7.64614634518936106257272316436, 8.526584282201241297832596046832, 9.249174739628621507905923356115

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