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Properties

Label 2-42e2-7.4-c3-0-30
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  + (3 + 5.19i)5-s + (−6 + 10.3i)11-s − 82·13-s + (−15 + 25.9i)17-s + (−34 − 58.8i)19-s + (108 + 187. i)23-s + (44.5 − 77.0i)25-s − 246·29-s + (56 − 96.9i)31-s + (−55 − 95.2i)37-s + 246·41-s − 172·43-s + (96 + 166. i)47-s + (279 − 483. i)53-s − 72·55-s + ⋯
L(s)  = 1  + (0.268 + 0.464i)5-s + (−0.164 + 0.284i)11-s − 1.74·13-s + (−0.214 + 0.370i)17-s + (−0.410 − 0.711i)19-s + (0.979 + 1.69i)23-s + (0.355 − 0.616i)25-s − 1.57·29-s + (0.324 − 0.561i)31-s + (−0.244 − 0.423i)37-s + 0.937·41-s − 0.609·43-s + (0.297 + 0.516i)47-s + (0.723 − 1.25i)53-s − 0.176·55-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

\(L(2)\) \(\approx\) \(1.306355275\)
\(L(\frac12)\) \(\approx\) \(1.306355275\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3 - 5.19i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (6 - 10.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 82T + 2.19e3T^{2} \)
17 \( 1 + (15 - 25.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (34 + 58.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-108 - 187. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 246T + 2.43e4T^{2} \)
31 \( 1 + (-56 + 96.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (55 + 95.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 246T + 6.89e4T^{2} \)
43 \( 1 + 172T + 7.95e4T^{2} \)
47 \( 1 + (-96 - 166. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-279 + 483. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-270 + 467. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (55 + 95.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (70 - 121. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 840T + 3.57e5T^{2} \)
73 \( 1 + (-275 + 476. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-104 - 180. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 516T + 5.71e5T^{2} \)
89 \( 1 + (699 + 1.21e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.58e3T + 9.12e5T^{2} \)
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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

−9.020422165924869893164370914615, −7.82018216569863322577176746571, −7.26146181736350600957438556331, −6.56478314621434743514068378396, −5.46909049364320176268653737552, −4.83991701422585894997080783907, −3.74940038934871434895651144395, −2.64690620064750497691621759005, −1.94108027749357494432653834038, −0.34269278439099835618194304101, 0.802563063807657712302807523489, 2.12033042530444823069714710582, 2.93537256464650623688272063301, 4.23005774667191642212710346574, 5.01126581062462448500396139857, 5.66017500480244854917819616112, 6.81063842696461118897875355416, 7.37959852258347167941793365177, 8.375699390839973244842821999413, 9.052221382273805301408406346154

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