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Properties

Label 2-42e2-63.41-c1-0-14
Degree $2$
Conductor $1764$
Sign $-0.0837 - 0.996i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.647 + 1.60i)3-s + (0.349 − 0.605i)5-s + (−2.16 + 2.08i)9-s + (−0.229 + 0.132i)11-s + (1.13 + 0.657i)13-s + (1.19 + 0.169i)15-s + 3.72·17-s + 0.441i·19-s + (4.29 + 2.48i)23-s + (2.25 + 3.90i)25-s + (−4.74 − 2.12i)27-s + (−0.273 + 0.157i)29-s + (−4.85 − 2.80i)31-s + (−0.361 − 0.283i)33-s + 0.702·37-s + ⋯
L(s)  = 1  + (0.373 + 0.927i)3-s + (0.156 − 0.270i)5-s + (−0.720 + 0.693i)9-s + (−0.0692 + 0.0399i)11-s + (0.315 + 0.182i)13-s + (0.309 + 0.0437i)15-s + 0.904·17-s + 0.101i·19-s + (0.896 + 0.517i)23-s + (0.451 + 0.781i)25-s + (−0.912 − 0.408i)27-s + (−0.0507 + 0.0292i)29-s + (−0.872 − 0.503i)31-s + (−0.0629 − 0.0492i)33-s + 0.115·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0837 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0837 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0837 - 0.996i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.0837 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.897417028\)
\(L(\frac12)\) \(\approx\) \(1.897417028\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.647 - 1.60i)T \)
7 \( 1 \)
good5 \( 1 + (-0.349 + 0.605i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.229 - 0.132i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.13 - 0.657i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.72T + 17T^{2} \)
19 \( 1 - 0.441iT - 19T^{2} \)
23 \( 1 + (-4.29 - 2.48i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.273 - 0.157i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.85 + 2.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.702T + 37T^{2} \)
41 \( 1 + (5.39 - 9.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.73 - 6.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.50 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.83iT - 53T^{2} \)
59 \( 1 + (-6.73 + 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.89 - 2.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.97 + 5.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 + 7.69iT - 73T^{2} \)
79 \( 1 + (0.698 + 1.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.72 + 6.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 + (9.18 - 5.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.464737764981459080383891227810, −8.915320611262176736939607054358, −8.027619308077831782124040199563, −7.32491972528540534604843634808, −6.08556012244687660830711283967, −5.32845372811390673339299582863, −4.58035280033828394073622748135, −3.58450750771166909975421801581, −2.82753447649261733988519144999, −1.39505939935566666834912265659, 0.72189149833520894356842133872, 1.99266543294992970144531562181, 2.96310092350404563616786529132, 3.81286560931759841157322326358, 5.23185878592378959013833382206, 5.92857164705025927126202044749, 6.96575385769647438164907364595, 7.28438786061439473195383424070, 8.478916895057997200650400069350, 8.739079025512980980592609190748

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