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Properties

Label 2-42e2-21.20-c3-0-19
Degree $2$
Conductor $1764$
Sign $0.970 - 0.239i$
Analytic cond. $104.079$
Root an. cond. $10.2019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.82·5-s − 58.3i·11-s − 38.5i·13-s + 32.2·17-s + 124. i·19-s + 201. i·23-s − 78.4·25-s + 104. i·29-s + 277. i·31-s − 47.6·37-s + 387.·41-s + 272.·43-s − 163.·47-s − 362. i·53-s − 398. i·55-s + ⋯
L(s)  = 1  + 0.610·5-s − 1.59i·11-s − 0.822i·13-s + 0.460·17-s + 1.50i·19-s + 1.82i·23-s − 0.627·25-s + 0.668i·29-s + 1.61i·31-s − 0.211·37-s + 1.47·41-s + 0.966·43-s − 0.506·47-s − 0.939i·53-s − 0.976i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.970 - 0.239i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ 0.970 - 0.239i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.409979436\)
\(L(\frac12)\) \(\approx\) \(2.409979436\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 - 6.82T + 125T^{2} \)
11 \( 1 + 58.3iT - 1.33e3T^{2} \)
13 \( 1 + 38.5iT - 2.19e3T^{2} \)
17 \( 1 - 32.2T + 4.91e3T^{2} \)
19 \( 1 - 124. iT - 6.85e3T^{2} \)
23 \( 1 - 201. iT - 1.21e4T^{2} \)
29 \( 1 - 104. iT - 2.43e4T^{2} \)
31 \( 1 - 277. iT - 2.97e4T^{2} \)
37 \( 1 + 47.6T + 5.06e4T^{2} \)
41 \( 1 - 387.T + 6.89e4T^{2} \)
43 \( 1 - 272.T + 7.95e4T^{2} \)
47 \( 1 + 163.T + 1.03e5T^{2} \)
53 \( 1 + 362. iT - 1.48e5T^{2} \)
59 \( 1 + 211.T + 2.05e5T^{2} \)
61 \( 1 + 234. iT - 2.26e5T^{2} \)
67 \( 1 - 524.T + 3.00e5T^{2} \)
71 \( 1 + 348. iT - 3.57e5T^{2} \)
73 \( 1 + 537. iT - 3.89e5T^{2} \)
79 \( 1 - 725.T + 4.93e5T^{2} \)
83 \( 1 - 392.T + 5.71e5T^{2} \)
89 \( 1 - 860.T + 7.04e5T^{2} \)
97 \( 1 - 978. iT - 9.12e5T^{2} \)
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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

−9.011260012417312186095555193378, −8.094923235541618396442732253203, −7.60488996702076097157217475134, −6.32512492542172673954267530226, −5.71293872535544135640375930444, −5.21212738389463701404728336481, −3.60706358416511526314574396416, −3.23458606680042777797858157749, −1.79764882588534346391725491270, −0.847096123049882555604133647282, 0.64858871217544084290565973881, 2.08597388767891249591130773469, 2.53200490429256944000220852958, 4.23041653863103412888984034504, 4.58615440964772486299296112078, 5.75016330444033999581809696123, 6.55551632383087180764252525455, 7.24914866937132367257509257372, 8.049714135739986820187663890792, 9.252428968278209040033262385805

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