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Properties

Label 2-42e2-12.11-c1-0-33
Degree $2$
Conductor $1764$
Sign $0.916 + 0.399i$
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  + (0.892 − 1.09i)2-s + (−0.406 − 1.95i)4-s + 2.56i·5-s + (−2.51 − 1.30i)8-s + (2.81 + 2.28i)10-s + 1.15·11-s + 0.578·13-s + (−3.66 + 1.59i)16-s + 5.39i·17-s − 6.20i·19-s + (5.02 − 1.04i)20-s + (1.02 − 1.26i)22-s + 7.62·23-s − 1.57·25-s + (0.516 − 0.634i)26-s + ⋯
L(s)  = 1  + (0.631 − 0.775i)2-s + (−0.203 − 0.979i)4-s + 1.14i·5-s + (−0.887 − 0.460i)8-s + (0.889 + 0.723i)10-s + 0.346·11-s + 0.160·13-s + (−0.917 + 0.398i)16-s + 1.30i·17-s − 1.42i·19-s + (1.12 − 0.233i)20-s + (0.218 − 0.269i)22-s + 1.58·23-s − 0.315·25-s + (0.101 − 0.124i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.916 + 0.399i$
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: \((2,\ 1764,\ (\ :1/2),\ 0.916 + 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.393293848\)
\(L(\frac12)\) \(\approx\) \(2.393293848\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.892 + 1.09i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2.56iT - 5T^{2} \)
11 \( 1 - 1.15T + 11T^{2} \)
13 \( 1 - 0.578T + 13T^{2} \)
17 \( 1 - 5.39iT - 17T^{2} \)
19 \( 1 + 6.20iT - 19T^{2} \)
23 \( 1 - 7.62T + 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 - 5.04iT - 31T^{2} \)
37 \( 1 - 9.83T + 37T^{2} \)
41 \( 1 + 6.21iT - 41T^{2} \)
43 \( 1 - 11.2iT - 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 - 4.53iT - 53T^{2} \)
59 \( 1 - 4.83T + 59T^{2} \)
61 \( 1 + 0.951T + 61T^{2} \)
67 \( 1 + 2.78iT - 67T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
79 \( 1 + 12.8iT - 79T^{2} \)
83 \( 1 - 8.77T + 83T^{2} \)
89 \( 1 - 5.68iT - 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−9.306191034937898448715713636279, −8.783532241369326966204520090150, −7.43037966893642460208945429908, −6.65636162273574337801565520027, −6.07691916515653514695257400368, −4.99615416192579118320013058955, −4.11207804723704980144582420289, −3.14325096268738434845464684115, −2.52471357162778011318777941178, −1.14991667179398320719532754293, 0.897890499083998098212942143357, 2.56436412334731509751620966437, 3.75877128552075967463990385093, 4.53932225088832121769749291449, 5.29654247728813967150585025531, 5.95699075536893009307571924969, 6.98076576303959053930035950263, 7.69855432257724613828983650389, 8.504989080111141212454609925304, 9.130687025920531372799956615143

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