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Properties

Label 2-42e2-7.2-c3-0-24
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  + (−9 + 15.5i)5-s + (18 + 31.1i)11-s − 10·13-s + (9 + 15.5i)17-s + (50 − 86.6i)19-s + (36 − 62.3i)23-s + (−99.5 − 172. i)25-s + 234·29-s + (8 + 13.8i)31-s + (113 − 195. i)37-s − 90·41-s + 452·43-s + (216 − 374. i)47-s + (207 + 358. i)53-s − 648·55-s + ⋯
L(s)  = 1  + (−0.804 + 1.39i)5-s + (0.493 + 0.854i)11-s − 0.213·13-s + (0.128 + 0.222i)17-s + (0.603 − 1.04i)19-s + (0.326 − 0.565i)23-s + (−0.796 − 1.37i)25-s + 1.49·29-s + (0.0463 + 0.0802i)31-s + (0.502 − 0.869i)37-s − 0.342·41-s + 1.60·43-s + (0.670 − 1.16i)47-s + (0.536 + 0.929i)53-s − 1.58·55-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: $\chi_{1764} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.926060383\)
\(L(\frac12)\) \(\approx\) \(1.926060383\)
\(L(\frac{5}{2})\) 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 \( 1 + (9 - 15.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-18 - 31.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 10T + 2.19e3T^{2} \)
17 \( 1 + (-9 - 15.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-50 + 86.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-36 + 62.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 234T + 2.43e4T^{2} \)
31 \( 1 + (-8 - 13.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-113 + 195. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 90T + 6.89e4T^{2} \)
43 \( 1 - 452T + 7.95e4T^{2} \)
47 \( 1 + (-216 + 374. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-207 - 358. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (342 + 592. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (211 - 365. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (166 + 287. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 360T + 3.57e5T^{2} \)
73 \( 1 + (13 + 22.5i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (256 - 443. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.18e3T + 5.71e5T^{2} \)
89 \( 1 + (315 - 545. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−9.100784485014251507452589711836, −8.119459920017407908967206283015, −7.20882021205489491860324162914, −6.96250339490282058935629204592, −6.03717887113895855828917687679, −4.77525633506588173175191203784, −4.02541703105511403385011242021, −3.03543768132552600439369260383, −2.31339744177336011169806873656, −0.71064849591796694813613207809, 0.69265111181377149649628938626, 1.32217538116967543484388056472, 2.96498771879852714710953592301, 3.91211950701074166263813684002, 4.65806581260244341721092494583, 5.48842906634384018100220404469, 6.30860509886872680089613247630, 7.52798203144024547504946464917, 8.056093602934370311458087087411, 8.810075573587609535778008108965

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