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Properties

Label 2-42e2-63.58-c1-0-36
Degree $2$
Conductor $1764$
Sign $-0.888 - 0.458i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (−1.5 − 2.59i)5-s + (1.5 − 2.59i)9-s + (−1.5 + 2.59i)11-s + (0.5 − 0.866i)13-s + (4.5 + 2.59i)15-s + (−3 − 5.19i)17-s + (2 − 3.46i)19-s + (1.5 + 2.59i)23-s + (−2 + 3.46i)25-s + 5.19i·27-s + (−1.5 − 2.59i)29-s + 5·31-s − 5.19i·33-s + (−1 + 1.73i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (−0.670 − 1.16i)5-s + (0.5 − 0.866i)9-s + (−0.452 + 0.783i)11-s + (0.138 − 0.240i)13-s + (1.16 + 0.670i)15-s + (−0.727 − 1.26i)17-s + (0.458 − 0.794i)19-s + (0.312 + 0.541i)23-s + (−0.400 + 0.692i)25-s + 0.999i·27-s + (−0.278 − 0.482i)29-s + 0.898·31-s − 0.904i·33-s + (−0.164 + 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.888 - 0.458i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9T + 47T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.5 + 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.101855695213267562430744635797, −8.023107721188545040004983287824, −7.27329039250474156054393421384, −6.40510794061928269552414959620, −5.19806384701196364574034383200, −4.84465189200982421524283435252, −4.15964326124540515954882731612, −2.87197920096424858712865534706, −1.15256704301415170785194348794, 0, 1.67276951697744958362217826515, 2.95576885973382889481795086449, 3.86720799093611103864298213487, 4.92602553551927137516175232335, 6.04488731330097663115359366291, 6.44889186425246614334859519269, 7.32775203713671477599203026669, 7.976005220292133554448482449198, 8.754185076815892050607934692683

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