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Properties

Label 2-42e2-63.41-c1-0-12
Degree $2$
Conductor $1764$
Sign $0.974 - 0.225i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.974 - 0.225i$
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: \((2,\ 1764,\ (\ :1/2),\ 0.974 - 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.173102076\)
\(L(\frac12)\) \(\approx\) \(1.173102076\)
\(L(\frac{3}{2})\) 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

−9.140665464525704418633302926387, −8.512198354112311609657478689452, −7.45651312989231826132790488544, −7.10455007860246815518293636029, −6.21503113871603099977409794426, −5.37582027834431970927237621320, −4.50826376383057038683240527298, −3.15760227400334112561689856323, −2.27598053506166116441107281984, −0.984960233914116200443141338435, 0.57864151116930339058354825044, 2.40000468958109875496058194539, 3.44555563505078094313807683232, 4.60174802981956246735231906730, 4.81727532985480364865260986088, 6.08904424365674199518751149604, 6.64728295356302983989937581351, 7.84133526695714414484531249061, 8.653970741695592453201027378661, 9.303798812544705107055811384863

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