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Properties

Label 2-42e2-12.11-c1-0-24
Degree $2$
Conductor $1764$
Sign $0.175 - 0.984i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.175 - 0.984i$
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: \((2,\ 1764,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.225083519\)
\(L(\frac12)\) \(\approx\) \(1.225083519\)
\(L(\frac{3}{2})\) 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

−9.190692511427642393196786509619, −8.645559123023217485297652353811, −7.912779209097338838995268717522, −6.96972681441137477532622508242, −6.54034672590597271873038543491, −5.45484511548460352066083367742, −4.84432766828249515781041583869, −3.72390374587355256790708311410, −2.47037477309501704144717398897, −0.913777484020455147920030241309, 0.73651442347996414876479456447, 1.96296669641487703332882896907, 3.05140546922542485398051715702, 3.87798945503418794225129085358, 4.95876133168806684301456782078, 5.60534690362082951586433665288, 7.01370128279261415442857385473, 7.70850975747832151036133888574, 8.653719379363270710135707747929, 9.005519984470748344020290291828

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