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Properties

Label 2-42e2-63.16-c1-0-4
Degree $2$
Conductor $1764$
Sign $-0.963 - 0.267i$
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  + (0.473 + 1.66i)3-s + 1.90·5-s + (−2.55 + 1.57i)9-s − 3.06·11-s + (−1.13 − 1.96i)13-s + (0.901 + 3.17i)15-s + (0.713 + 1.23i)17-s + (−2.98 + 5.16i)19-s − 7.15·23-s − 1.37·25-s + (−3.83 − 3.50i)27-s + (0.468 − 0.810i)29-s + (−4.11 + 7.11i)31-s + (−1.45 − 5.10i)33-s + (−1.41 + 2.45i)37-s + ⋯
L(s)  = 1  + (0.273 + 0.961i)3-s + 0.851·5-s + (−0.850 + 0.526i)9-s − 0.924·11-s + (−0.313 − 0.543i)13-s + (0.232 + 0.818i)15-s + (0.173 + 0.299i)17-s + (−0.684 + 1.18i)19-s − 1.49·23-s − 0.275·25-s + (−0.738 − 0.674i)27-s + (0.0869 − 0.150i)29-s + (−0.738 + 1.27i)31-s + (−0.252 − 0.889i)33-s + (−0.232 + 0.403i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.963 - 0.267i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.963 - 0.267i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.102096975\)
\(L(\frac12)\) \(\approx\) \(1.102096975\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.473 - 1.66i)T \)
7 \( 1 \)
good5 \( 1 - 1.90T + 5T^{2} \)
11 \( 1 + 3.06T + 11T^{2} \)
13 \( 1 + (1.13 + 1.96i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.713 - 1.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.98 - 5.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.15T + 23T^{2} \)
29 \( 1 + (-0.468 + 0.810i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.11 - 7.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.41 - 2.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.31 - 9.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.98 + 5.16i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.483 + 0.837i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.45 - 9.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.68 - 9.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.449 - 0.778i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.813 - 1.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.36T + 71T^{2} \)
73 \( 1 + (-0.996 - 1.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.16 - 7.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.98 + 13.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.58 + 4.48i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.922 - 1.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.854511042861858656031327587916, −8.942609980157379414576996220255, −8.171184831095651858356743921234, −7.54309538708185229701088508925, −6.00703826978846762021405412471, −5.73758649148308060582605166969, −4.74438994583461113459376765227, −3.82306469724441868759445995686, −2.80369447178129606550950928880, −1.88005116728328072058197112503, 0.35010501669598161429840818098, 2.10490832361259835569796846959, 2.34505788979854926992302462283, 3.76817791967911633487607016495, 5.02362675915125989521480061107, 5.87395201505665088748824704899, 6.50007030394600489362096081084, 7.44867232834097596291459974103, 7.978888512066807362005798882492, 9.018880539216452019717987574793

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