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Properties

Label 2-42e2-12.11-c1-0-61
Degree $2$
Conductor $1764$
Sign $0.832 + 0.554i$
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  + (1.39 − 0.258i)2-s + (1.86 − 0.719i)4-s + 0.732i·5-s + (2.40 − 1.48i)8-s + (0.189 + 1.01i)10-s + 2.03·11-s − 1.41·13-s + (2.96 − 2.68i)16-s − 4.19i·17-s − 5.56i·19-s + (0.526 + 1.36i)20-s + (2.83 − 0.526i)22-s + 2.03·23-s + 4.46·25-s + (−1.96 + 0.366i)26-s + ⋯
L(s)  = 1  + (0.983 − 0.183i)2-s + (0.933 − 0.359i)4-s + 0.327i·5-s + (0.851 − 0.524i)8-s + (0.0599 + 0.321i)10-s + 0.613·11-s − 0.392·13-s + (0.741 − 0.671i)16-s − 1.01i·17-s − 1.27i·19-s + (0.117 + 0.305i)20-s + (0.603 − 0.112i)22-s + 0.424·23-s + 0.892·25-s + (−0.385 + 0.0717i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.525397517\)
\(L(\frac12)\) \(\approx\) \(3.525397517\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.258i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 0.732iT - 5T^{2} \)
11 \( 1 - 2.03T + 11T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + 4.19iT - 17T^{2} \)
19 \( 1 + 5.56iT - 19T^{2} \)
23 \( 1 - 2.03T + 23T^{2} \)
29 \( 1 + 5.27iT - 29T^{2} \)
31 \( 1 - 9.63iT - 31T^{2} \)
37 \( 1 - 3.46T + 37T^{2} \)
41 \( 1 - 8.73iT - 41T^{2} \)
43 \( 1 - 7.86iT - 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 9.14iT - 53T^{2} \)
59 \( 1 + 4.98T + 59T^{2} \)
61 \( 1 - 9.41T + 61T^{2} \)
67 \( 1 - 2.87iT - 67T^{2} \)
71 \( 1 + 9.08T + 71T^{2} \)
73 \( 1 + 1.69T + 73T^{2} \)
79 \( 1 - 4.98iT - 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + 8.73iT - 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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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

−9.404401541434643687039780866097, −8.410384663058786260512482760462, −7.12626757639907555020433279881, −6.94249792118003183731152579533, −5.94431029829219901594176372890, −4.92604620695517773929279887576, −4.41098780031328127538138394307, −3.12502295476566481424503319724, −2.58320271051344300345725209613, −1.09318554584262477602384332641, 1.40937871500138743828337130498, 2.53248836757781155476504982751, 3.76565627320591944525394026394, 4.26219786641708679732427642508, 5.41259801459151930381302383951, 5.95026328119400791738251536236, 6.89665151817987588903031076119, 7.61512478808423313931554933869, 8.477634474435259521856232097603, 9.270172121968113667793398415090

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