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Properties

Label 2-42e2-12.11-c1-0-18
Degree $2$
Conductor $1764$
Sign $-0.986 - 0.162i$
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  + (0.545 + 1.30i)2-s + (−1.40 + 1.42i)4-s + 0.698i·5-s + (−2.62 − 1.05i)8-s + (−0.911 + 0.380i)10-s + 2.55·11-s + 1.88·13-s + (−0.0532 − 3.99i)16-s + 3.97i·17-s + 7.05i·19-s + (−0.994 − 0.980i)20-s + (1.39 + 3.32i)22-s − 4.02·23-s + 4.51·25-s + (1.02 + 2.45i)26-s + ⋯
L(s)  = 1  + (0.385 + 0.922i)2-s + (−0.702 + 0.711i)4-s + 0.312i·5-s + (−0.927 − 0.373i)8-s + (−0.288 + 0.120i)10-s + 0.769·11-s + 0.521·13-s + (−0.0133 − 0.999i)16-s + 0.963i·17-s + 1.61i·19-s + (−0.222 − 0.219i)20-s + (0.296 + 0.709i)22-s − 0.839·23-s + 0.902·25-s + (0.201 + 0.481i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.986 - 0.162i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.986 - 0.162i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.599927203\)
\(L(\frac12)\) \(\approx\) \(1.599927203\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.545 - 1.30i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 0.698iT - 5T^{2} \)
11 \( 1 - 2.55T + 11T^{2} \)
13 \( 1 - 1.88T + 13T^{2} \)
17 \( 1 - 3.97iT - 17T^{2} \)
19 \( 1 - 7.05iT - 19T^{2} \)
23 \( 1 + 4.02T + 23T^{2} \)
29 \( 1 + 1.86iT - 29T^{2} \)
31 \( 1 + 0.941iT - 31T^{2} \)
37 \( 1 + 7.49T + 37T^{2} \)
41 \( 1 - 10.6iT - 41T^{2} \)
43 \( 1 - 3.97iT - 43T^{2} \)
47 \( 1 + 8.90T + 47T^{2} \)
53 \( 1 - 0.529iT - 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 2.72iT - 67T^{2} \)
71 \( 1 - 3.51T + 71T^{2} \)
73 \( 1 + 2.75T + 73T^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 - 11.8iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−9.596790142919807973708394458219, −8.514991517022919588170973497737, −8.159586856723346300130899356809, −7.20632730008443186184494407684, −6.20979499237111625424045653606, −6.04726076558238325263859770568, −4.77734766858561609826559053637, −3.89306887641402042165394804112, −3.23759119131882242722897520783, −1.57463346306443153012111233324, 0.54484806619979619644088151832, 1.75667253381758288250559976270, 2.90479497768704917494593407696, 3.79371204975777214172372551489, 4.73780199456350812563520467992, 5.35853622129617313214711485129, 6.45800232797327716016093038934, 7.19088706822089324785008394402, 8.642125894506200310806066119913, 8.918968360955407171978829063067

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