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Properties

Label 2-42e2-7.4-c3-0-32
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  + (−5.29 − 9.16i)5-s + (27.6 − 47.8i)11-s + 83.5·13-s + (−47.6 + 82.4i)17-s + (−41.7 − 72.3i)19-s + (82.8 + 143. i)23-s + (6.5 − 11.2i)25-s + 110.·29-s + (41.7 − 72.3i)31-s + (−39 − 67.5i)37-s + 412.·41-s + 148·43-s + (232. + 403. i)47-s + (−55.2 + 95.6i)53-s − 584.·55-s + ⋯
L(s)  = 1  + (−0.473 − 0.819i)5-s + (0.757 − 1.31i)11-s + 1.78·13-s + (−0.679 + 1.17i)17-s + (−0.504 − 0.873i)19-s + (0.751 + 1.30i)23-s + (0.0520 − 0.0900i)25-s + 0.707·29-s + (0.241 − 0.419i)31-s + (−0.173 − 0.300i)37-s + 1.57·41-s + 0.524·43-s + (0.722 + 1.25i)47-s + (−0.143 + 0.247i)53-s − 1.43·55-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

\(L(2)\) \(\approx\) \(2.371800868\)
\(L(\frac12)\) \(\approx\) \(2.371800868\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.29 + 9.16i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-27.6 + 47.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 83.5T + 2.19e3T^{2} \)
17 \( 1 + (47.6 - 82.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (41.7 + 72.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-82.8 - 143. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 110.T + 2.43e4T^{2} \)
31 \( 1 + (-41.7 + 72.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (39 + 67.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 412.T + 6.89e4T^{2} \)
43 \( 1 - 148T + 7.95e4T^{2} \)
47 \( 1 + (-232. - 403. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (55.2 - 95.6i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (275. - 476. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-292. - 506. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-130 + 225. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 718.T + 3.57e5T^{2} \)
73 \( 1 + (334. - 578. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (332 + 575. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 126.T + 5.71e5T^{2} \)
89 \( 1 + (439. + 760. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.16e3T + 9.12e5T^{2} \)
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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

−8.768038598961307817183022046785, −8.357112047580786523887379300811, −7.29530672822287897994638022524, −6.08363281570929441159600003867, −5.92511674118907006281846061069, −4.43368270056001419763765743714, −3.96209974165368010061610806824, −2.95817950311133866848011260498, −1.36001185679994170545148623977, −0.72927342379424361094395929349, 0.864645094989009620931127662709, 2.08996894380896648122877870523, 3.15734354982639115983192019874, 4.05754773588938849577952818119, 4.75348323626285370680715505864, 6.09480160990345751021926612392, 6.75351716071401313501572226310, 7.25910890693175565756091603033, 8.352291630525062491683529977237, 8.963902254511645208551423582460

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