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Properties

Label 2-42e2-7.4-c3-0-8
Degree $2$
Conductor $1764$
Sign $0.701 - 0.712i$
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  + (−7 − 12.1i)5-s + (2 − 3.46i)11-s − 54·13-s + (7 − 12.1i)17-s + (46 + 79.6i)19-s + (−76 − 131. i)23-s + (−35.5 + 61.4i)25-s + 106·29-s + (−72 + 124. i)31-s + (−79 − 136. i)37-s − 390·41-s − 508·43-s + (264 + 457. i)47-s + (303 − 524. i)53-s − 56·55-s + ⋯
L(s)  = 1  + (−0.626 − 1.08i)5-s + (0.0548 − 0.0949i)11-s − 1.15·13-s + (0.0998 − 0.172i)17-s + (0.555 + 0.962i)19-s + (−0.689 − 1.19i)23-s + (−0.284 + 0.491i)25-s + 0.678·29-s + (−0.417 + 0.722i)31-s + (−0.351 − 0.607i)37-s − 1.48·41-s − 1.80·43-s + (0.819 + 1.41i)47-s + (0.785 − 1.36i)53-s − 0.137·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.701 - 0.712i$
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: \((2,\ 1764,\ (\ :3/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8703959735\)
\(L(\frac12)\) \(\approx\) \(0.8703959735\)
\(L(\frac{5}{2})\) 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 \( 1 + (7 + 12.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 54T + 2.19e3T^{2} \)
17 \( 1 + (-7 + 12.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-46 - 79.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (76 + 131. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 106T + 2.43e4T^{2} \)
31 \( 1 + (72 - 124. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (79 + 136. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 390T + 6.89e4T^{2} \)
43 \( 1 + 508T + 7.95e4T^{2} \)
47 \( 1 + (-264 - 457. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-303 + 524. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-182 + 315. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-339 - 587. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (422 - 730. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 8T + 3.57e5T^{2} \)
73 \( 1 + (211 - 365. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (192 + 332. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 548T + 5.71e5T^{2} \)
89 \( 1 + (597 + 1.03e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.50e3T + 9.12e5T^{2} \)
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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

−8.745081393579294336469340450899, −8.430390731038530503622442267065, −7.53626397904229880616780520137, −6.77634702296945732052445013266, −5.61477844296193131685849828532, −4.89486061209085022996045719263, −4.19399119942902712756082169394, −3.16002931958306381153882573047, −1.90808326946200497236767025173, −0.73014116373006961153396147152, 0.25575609832500051718094835141, 1.88359291103016062084201304887, 2.94720772149283709558080133627, 3.62416465456978357378464029709, 4.72353801754445482187896605658, 5.56215768741735501964362194228, 6.73881941205815215816568921045, 7.18682960306889698664887081531, 7.86463778025056962124715734647, 8.803236541659455994071232772075

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