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Properties

Label 2-42e2-63.59-c1-0-17
Degree $2$
Conductor $1764$
Sign $0.610 - 0.791i$
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  + (1.60 + 0.649i)3-s + 2.96·5-s + (2.15 + 2.08i)9-s + 4.72i·11-s + (−3.54 − 2.04i)13-s + (4.76 + 1.92i)15-s + (−0.835 + 1.44i)17-s + (4.25 − 2.45i)19-s + 4.91i·23-s + 3.82·25-s + (2.11 + 4.74i)27-s + (0.238 − 0.137i)29-s + (1.38 − 0.801i)31-s + (−3.06 + 7.59i)33-s + (−1.69 − 2.93i)37-s + ⋯
L(s)  = 1  + (0.927 + 0.374i)3-s + 1.32·5-s + (0.719 + 0.694i)9-s + 1.42i·11-s + (−0.981 − 0.566i)13-s + (1.23 + 0.497i)15-s + (−0.202 + 0.350i)17-s + (0.975 − 0.563i)19-s + 1.02i·23-s + 0.764·25-s + (0.406 + 0.913i)27-s + (0.0442 − 0.0255i)29-s + (0.249 − 0.143i)31-s + (−0.534 + 1.32i)33-s + (−0.278 − 0.483i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.610 - 0.791i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.610 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.040423248\)
\(L(\frac12)\) \(\approx\) \(3.040423248\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 - 0.649i)T \)
7 \( 1 \)
good5 \( 1 - 2.96T + 5T^{2} \)
11 \( 1 - 4.72iT - 11T^{2} \)
13 \( 1 + (3.54 + 2.04i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.835 - 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.25 + 2.45i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.91iT - 23T^{2} \)
29 \( 1 + (-0.238 + 0.137i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.38 + 0.801i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.55 - 6.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.22 - 9.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.49 + 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.707 + 0.408i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.37 + 2.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.23 - 3.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.80 + 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (13.6 + 7.88i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.15 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.03 + 6.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.60 - 7.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.00 + 4.04i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.484423314022896881667121537194, −8.935744143911182042027750957898, −7.64628074351666383168182006371, −7.33575776713422741187025487498, −6.17982465446473824738720230751, −5.13410007422740422646710622988, −4.62861133714079415313632357208, −3.29862961148389039187357003935, −2.37378015043624818822933619108, −1.65512708367961573845053661655, 1.06880625888629025397240322000, 2.26308408451702716430602357039, 2.89703831850115924840615085547, 4.03481554100416864422779118873, 5.25674880293715701746386183677, 6.01285112794386020471053031428, 6.83836029584171200306314881338, 7.58605442725885891229831012209, 8.674238683610501255306545019816, 9.006257501655152619241610379484

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