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Properties

Label 2-12e3-144.83-c1-0-11
Degree $2$
Conductor $1728$
Sign $0.999 - 0.0252i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
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  + (−0.323 + 1.20i)5-s + (−0.140 + 0.242i)7-s + (−0.823 − 3.07i)11-s + (−0.740 + 2.76i)13-s − 3.72i·17-s + (4.10 − 4.10i)19-s + (−1.57 + 0.909i)23-s + (2.98 + 1.72i)25-s + (1.02 + 3.83i)29-s + (8.81 − 5.08i)31-s + (−0.247 − 0.247i)35-s + (1.76 − 1.76i)37-s + (2.66 + 4.62i)41-s + (−6.89 + 1.84i)43-s + (5.48 − 9.50i)47-s + ⋯
L(s)  = 1  + (−0.144 + 0.539i)5-s + (−0.0530 + 0.0918i)7-s + (−0.248 − 0.926i)11-s + (−0.205 + 0.766i)13-s − 0.902i·17-s + (0.942 − 0.942i)19-s + (−0.328 + 0.189i)23-s + (0.596 + 0.344i)25-s + (0.190 + 0.711i)29-s + (1.58 − 0.913i)31-s + (−0.0418 − 0.0418i)35-s + (0.290 − 0.290i)37-s + (0.416 + 0.721i)41-s + (−1.05 + 0.281i)43-s + (0.800 − 1.38i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.999 - 0.0252i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.999 - 0.0252i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.634538474\)
\(L(\frac12)\) \(\approx\) \(1.634538474\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (0.323 - 1.20i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.140 - 0.242i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.823 + 3.07i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.740 - 2.76i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + 3.72iT - 17T^{2} \)
19 \( 1 + (-4.10 + 4.10i)T - 19iT^{2} \)
23 \( 1 + (1.57 - 0.909i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.02 - 3.83i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-8.81 + 5.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.76 + 1.76i)T - 37iT^{2} \)
41 \( 1 + (-2.66 - 4.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.89 - 1.84i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-5.48 + 9.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.58 - 8.58i)T + 53iT^{2} \)
59 \( 1 + (-5.38 - 1.44i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.23 - 1.66i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.75 + 1.00i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.6iT - 71T^{2} \)
73 \( 1 + 9.30iT - 73T^{2} \)
79 \( 1 + (-8.70 - 5.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.19 + 0.588i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.87T + 89T^{2} \)
97 \( 1 + (-9.19 + 15.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.264947337429048041659800546572, −8.647169840002819872469996951341, −7.59310676613266015139486712109, −7.01670379593171551416000263692, −6.16886460148764870641534084007, −5.24587164620789121379476824902, −4.36484351540047657666206165427, −3.17183802083380942613930391997, −2.55007292126832143593818791232, −0.858998788016689282222806109924, 0.938197753825330711659845544484, 2.23712048681477407565536127046, 3.41485422028535112089328150122, 4.41666924476058081840910350577, 5.16044912982974495667317945231, 6.05113119314616828691755585197, 6.98096990451328292154432148738, 7.939068634047387310859770718139, 8.325517690174112313759725319545, 9.363852778067130420790894888787

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