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Properties

Label 2-12e3-9.2-c2-0-5
Degree $2$
Conductor $1728$
Sign $-0.677 - 0.735i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.15 − 0.669i)5-s + (0.328 − 0.569i)7-s + (7.33 + 4.23i)11-s + (0.615 + 1.06i)13-s + 5.78i·17-s − 22.7·19-s + (−17.4 + 10.0i)23-s + (−11.6 + 20.0i)25-s + (−35.5 − 20.5i)29-s + (7.57 + 13.1i)31-s − 0.880i·35-s + 51.2·37-s + (−17.7 + 10.2i)41-s + (−3.14 + 5.44i)43-s + (−31.9 − 18.4i)47-s + ⋯
L(s)  = 1  + (0.231 − 0.133i)5-s + (0.0469 − 0.0813i)7-s + (0.667 + 0.385i)11-s + (0.0473 + 0.0820i)13-s + 0.340i·17-s − 1.19·19-s + (−0.759 + 0.438i)23-s + (−0.464 + 0.803i)25-s + (−1.22 − 0.707i)29-s + (0.244 + 0.423i)31-s − 0.0251i·35-s + 1.38·37-s + (−0.433 + 0.250i)41-s + (−0.0731 + 0.126i)43-s + (−0.679 − 0.392i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.677 - 0.735i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.677 - 0.735i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8775375234\)
\(L(\frac12)\) \(\approx\) \(0.8775375234\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-1.15 + 0.669i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-0.328 + 0.569i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-7.33 - 4.23i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-0.615 - 1.06i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 5.78iT - 289T^{2} \)
19 \( 1 + 22.7T + 361T^{2} \)
23 \( 1 + (17.4 - 10.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (35.5 + 20.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-7.57 - 13.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 51.2T + 1.36e3T^{2} \)
41 \( 1 + (17.7 - 10.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (3.14 - 5.44i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (31.9 + 18.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 85.9iT - 2.80e3T^{2} \)
59 \( 1 + (51.2 - 29.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-38.1 + 66.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-48.2 - 83.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 31.2iT - 5.04e3T^{2} \)
73 \( 1 + 24.9T + 5.32e3T^{2} \)
79 \( 1 + (67.4 - 116. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (98.6 + 56.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 135. iT - 7.92e3T^{2} \)
97 \( 1 + (68.8 - 119. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.600213186546450604580282076305, −8.604788540417137898425846074914, −7.927218356887296925436313170710, −6.98732558351878574230551850675, −6.21891392062135928857764163549, −5.45333059821856596981290873422, −4.32425717058922371558909731794, −3.71626097105741124969049464835, −2.30650466344235115547820946111, −1.41191034188247970076332982206, 0.22099152820205443602640796525, 1.69239243662994862507572769025, 2.69199394026881688037533502777, 3.86323775325497069497957960744, 4.58494248978596533476636689175, 5.83319607072369763172000038760, 6.28256316712497027201633856016, 7.23903318354130576533743912370, 8.143881230038484656014383825957, 8.837159582231721964034852162097

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