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Properties

Label 2-42e2-7.4-c3-0-25
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  + (−4 − 6.92i)5-s + (−20 + 34.6i)11-s − 12·13-s + (−29 + 50.2i)17-s + (−13 − 22.5i)19-s + (−32 − 55.4i)23-s + (30.5 − 52.8i)25-s + 62·29-s + (−126 + 218. i)31-s + (−13 − 22.5i)37-s − 6·41-s + 416·43-s + (−198 − 342. i)47-s + (−225 + 389. i)53-s + 320·55-s + ⋯
L(s)  = 1  + (−0.357 − 0.619i)5-s + (−0.548 + 0.949i)11-s − 0.256·13-s + (−0.413 + 0.716i)17-s + (−0.156 − 0.271i)19-s + (−0.290 − 0.502i)23-s + (0.244 − 0.422i)25-s + 0.397·29-s + (−0.730 + 1.26i)31-s + (−0.0577 − 0.100i)37-s − 0.0228·41-s + 1.47·43-s + (−0.614 − 1.06i)47-s + (−0.583 + 1.01i)53-s + 0.784·55-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

\(L(2)\) \(\approx\) \(1.316514793\)
\(L(\frac12)\) \(\approx\) \(1.316514793\)
\(L(\frac{5}{2})\) 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 \( 1 + (4 + 6.92i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (20 - 34.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 12T + 2.19e3T^{2} \)
17 \( 1 + (29 - 50.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (13 + 22.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (32 + 55.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 62T + 2.43e4T^{2} \)
31 \( 1 + (126 - 218. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (13 + 22.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 6T + 6.89e4T^{2} \)
43 \( 1 - 416T + 7.95e4T^{2} \)
47 \( 1 + (198 + 342. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (225 - 389. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-137 + 237. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-288 - 498. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-238 + 412. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 448T + 3.57e5T^{2} \)
73 \( 1 + (-79 + 136. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-468 - 810. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 530T + 5.71e5T^{2} \)
89 \( 1 + (195 + 337. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 214T + 9.12e5T^{2} \)
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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

−8.708742635729165716083432075491, −8.149202634123166378105677685223, −7.25059372457522655395115084363, −6.53547911758221244498087788488, −5.41894259085668671644672603423, −4.66937361502452860721835247861, −3.99438439144310755964852714049, −2.70212319534457241403283708049, −1.72746003118927468358898249699, −0.40899398155359271184388571442, 0.69162672813409337139823330084, 2.21778901966792836883252487132, 3.10214587634098671619135994730, 3.92430849165303129349854604060, 5.03051243112049013496082168126, 5.85540061850950779536574930553, 6.70060123993165709978008939452, 7.56403956505456797244465115002, 8.098082506055592239600883389068, 9.105236130410637616120455831463

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