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Properties

Label 2-42e2-7.4-c3-0-6
Degree $2$
Conductor $1764$
Sign $-0.605 - 0.795i$
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  + (−2 − 3.46i)5-s + (−10 + 17.3i)11-s + 4·13-s + (−12 + 20.7i)17-s + (22 + 38.1i)19-s + (36 + 62.3i)23-s + (54.5 − 94.3i)25-s + 38·29-s + (92 − 159. i)31-s + (15 + 25.9i)37-s − 216·41-s − 164·43-s + (−260 − 450. i)47-s + (−73 + 126. i)53-s + 80·55-s + ⋯
L(s)  = 1  + (−0.178 − 0.309i)5-s + (−0.274 + 0.474i)11-s + 0.0853·13-s + (−0.171 + 0.296i)17-s + (0.265 + 0.460i)19-s + (0.326 + 0.565i)23-s + (0.435 − 0.755i)25-s + 0.243·29-s + (0.533 − 0.923i)31-s + (0.0666 + 0.115i)37-s − 0.822·41-s − 0.581·43-s + (−0.806 − 1.39i)47-s + (−0.189 + 0.327i)53-s + 0.196·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8014906983\)
\(L(\frac12)\) \(\approx\) \(0.8014906983\)
\(L(\frac{5}{2})\) 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 \( 1 + (2 + 3.46i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (10 - 17.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 4T + 2.19e3T^{2} \)
17 \( 1 + (12 - 20.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-22 - 38.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-36 - 62.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 38T + 2.43e4T^{2} \)
31 \( 1 + (-92 + 159. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-15 - 25.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 216T + 6.89e4T^{2} \)
43 \( 1 + 164T + 7.95e4T^{2} \)
47 \( 1 + (260 + 450. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (73 - 126. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (230 - 398. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-314 - 543. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (278 - 481. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 592T + 3.57e5T^{2} \)
73 \( 1 + (-512 + 886. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-52 - 90.0i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 324T + 5.71e5T^{2} \)
89 \( 1 + (448 + 775. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 920T + 9.12e5T^{2} \)
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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

−9.146795587603413720654597412551, −8.406775288103347904102964578023, −7.71342670613578346275128236124, −6.86419916446445564615308252137, −6.00289942750689493643912683824, −5.07784735614588669937628326463, −4.32135133891677080351519333796, −3.33829678064035108361884927186, −2.23156081553992269578077556720, −1.11176387415633911350809346862, 0.18302262650158971970537391434, 1.41719066393844965239040771789, 2.77629112207414538387656546430, 3.39561290030609142901828588870, 4.63870743234786396276210762386, 5.29562113833009318241056819934, 6.42678431701514713838670882063, 6.96639465165704770090151683921, 7.934811107406342843475578495141, 8.608631575417185464697964357267

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