Properties

Label 2-42e2-28.27-c1-0-36
Degree $2$
Conductor $1764$
Sign $-0.0532 - 0.998i$
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  + (1.20 + 0.736i)2-s + (0.914 + 1.77i)4-s − 1.08i·5-s + (−0.207 + 2.82i)8-s + (0.797 − 1.30i)10-s + 2.08i·11-s − 2.61i·13-s + (−2.32 + 3.25i)16-s + 4.46i·17-s + 1.12·19-s + (1.92 − 0.989i)20-s + (−1.53 + 2.51i)22-s + 7.11i·23-s + 3.82·25-s + (1.92 − 3.15i)26-s + ⋯
L(s)  = 1  + (0.853 + 0.521i)2-s + (0.457 + 0.889i)4-s − 0.484i·5-s + (−0.0732 + 0.997i)8-s + (0.252 − 0.413i)10-s + 0.628i·11-s − 0.724i·13-s + (−0.582 + 0.813i)16-s + 1.08i·17-s + 0.258·19-s + (0.430 − 0.221i)20-s + (−0.327 + 0.536i)22-s + 1.48i·23-s + 0.765·25-s + (0.377 − 0.618i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0532 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0532 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0532 - 0.998i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.0532 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.784727802\)
\(L(\frac12)\) \(\approx\) \(2.784727802\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 - 0.736i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.08iT - 5T^{2} \)
11 \( 1 - 2.08iT - 11T^{2} \)
13 \( 1 + 2.61iT - 13T^{2} \)
17 \( 1 - 4.46iT - 17T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 - 7.11iT - 23T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
31 \( 1 - 7.70T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 5.54iT - 41T^{2} \)
43 \( 1 - 7.97iT - 43T^{2} \)
47 \( 1 + 5.44T + 47T^{2} \)
53 \( 1 - 6.48T + 53T^{2} \)
59 \( 1 - 8.82T + 59T^{2} \)
61 \( 1 + 13.0iT - 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 7.97iT - 73T^{2} \)
79 \( 1 - 4.16iT - 79T^{2} \)
83 \( 1 + 4.31T + 83T^{2} \)
89 \( 1 + 4.01iT - 89T^{2} \)
97 \( 1 + 3.82iT - 97T^{2} \)
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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

−9.449895407664677668069302927465, −8.336698606105846255184959147958, −7.980051435393125733061925752551, −6.99273343083420408815717210922, −6.23368751324836117908637829271, −5.33818323333279786023282493024, −4.72456886338819424066094880565, −3.75485839554832555740473414261, −2.85165886021713775276410631828, −1.50772180543232620050629272840, 0.816137085359958016218680752719, 2.34055035396452087219776500703, 3.01788151796780653131800738279, 4.06448331060873097856174062518, 4.88446907052598332170055460295, 5.74160783493281896014583467776, 6.74704027522326716629327022555, 7.05086807139249836081645834389, 8.457535346232197041670233614671, 9.166786769418562317537015947746

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