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Properties

Label 2-42e2-63.47-c1-0-23
Degree $2$
Conductor $1764$
Sign $0.964 + 0.262i$
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  + (1.31 − 1.12i)3-s − 0.0764·5-s + (0.462 − 2.96i)9-s + 5.38i·11-s + (4.60 − 2.65i)13-s + (−0.100 + 0.0860i)15-s + (1.89 + 3.27i)17-s + (4.33 + 2.50i)19-s + 2.33i·23-s − 4.99·25-s + (−2.73 − 4.42i)27-s + (8.84 + 5.10i)29-s + (−4.97 − 2.87i)31-s + (6.06 + 7.08i)33-s + (0.354 − 0.613i)37-s + ⋯
L(s)  = 1  + (0.759 − 0.650i)3-s − 0.0341·5-s + (0.154 − 0.988i)9-s + 1.62i·11-s + (1.27 − 0.737i)13-s + (−0.0259 + 0.0222i)15-s + (0.458 + 0.794i)17-s + (0.995 + 0.574i)19-s + 0.487i·23-s − 0.998·25-s + (−0.525 − 0.850i)27-s + (1.64 + 0.948i)29-s + (−0.893 − 0.516i)31-s + (1.05 + 1.23i)33-s + (0.0582 − 0.100i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.964 + 0.262i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.964 + 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.474106331\)
\(L(\frac12)\) \(\approx\) \(2.474106331\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 + 1.12i)T \)
7 \( 1 \)
good5 \( 1 + 0.0764T + 5T^{2} \)
11 \( 1 - 5.38iT - 11T^{2} \)
13 \( 1 + (-4.60 + 2.65i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.89 - 3.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.33 - 2.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.33iT - 23T^{2} \)
29 \( 1 + (-8.84 - 5.10i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.97 + 2.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.354 + 0.613i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.29 + 5.71i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.716 + 1.24i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.46 - 2.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.4 + 6.05i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.289 + 0.502i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.40 + 1.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.63 - 4.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.32iT - 71T^{2} \)
73 \( 1 + (-6.17 + 3.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.469 + 0.812i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.49 + 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.51 + 2.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.18 + 3.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.182051518752013163150223304811, −8.355502091688553101374228955830, −7.70575664706550296430526707606, −7.09048297062062929664384429825, −6.14431765675727507346475488296, −5.30518809782639742553577024308, −3.96206777316966802169462141118, −3.37332783658303991165401959836, −2.08846320168675023712863358426, −1.21936770867569920934780942685, 1.05835983387720801861702797864, 2.62517565310673530987684572252, 3.39830727829985801085349218354, 4.15645253253550974472331703291, 5.22394461687747791824536369769, 6.02985169679165481028940911342, 7.00935786270493071450244099747, 8.087786910591485841726681220519, 8.538095310201802508266206315143, 9.255686291968571622524519552948

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