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Properties

Label 2-12e3-72.13-c1-0-0
Degree $2$
Conductor $1728$
Sign $0.0871 - 0.996i$
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  + (−3.35 − 1.93i)5-s + (−1.93 − 3.35i)7-s + (−2.59 + 1.5i)11-s + (−3.35 − 1.93i)13-s − 2i·19-s + (1.93 − 3.35i)23-s + (5.00 + 8.66i)25-s + (3.35 − 1.93i)29-s + (−1.93 + 3.35i)31-s + 15.0i·35-s + 7.74i·37-s + (4.5 − 7.79i)41-s + (−6.06 + 3.5i)43-s + (1.93 + 3.35i)47-s + (−4.00 + 6.92i)49-s + ⋯
L(s)  = 1  + (−1.50 − 0.866i)5-s + (−0.731 − 1.26i)7-s + (−0.783 + 0.452i)11-s + (−0.930 − 0.537i)13-s − 0.458i·19-s + (0.403 − 0.699i)23-s + (1.00 + 1.73i)25-s + (0.622 − 0.359i)29-s + (−0.347 + 0.602i)31-s + 2.53i·35-s + 1.27i·37-s + (0.702 − 1.21i)41-s + (−0.924 + 0.533i)43-s + (0.282 + 0.489i)47-s + (−0.571 + 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.0871 - 0.996i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.0871 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09096277824\)
\(L(\frac12)\) \(\approx\) \(0.09096277824\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.35 + 1.93i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.93 + 3.35i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.35 + 1.93i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (-1.93 + 3.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.35 + 1.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.93 - 3.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.74iT - 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.06 - 3.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.93 - 3.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.74iT - 53T^{2} \)
59 \( 1 + (2.59 + 1.5i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.0 + 5.80i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.52 + 5.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.74T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + (-1.93 - 3.35i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.59 + 1.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.518977518182561945729915264257, −8.582531618824151820471855850483, −7.76325944771997219724321689532, −7.37313602328836520311103677273, −6.55779090403128772193650529045, −5.03234063646413588760325577711, −4.60742414328669124976555884631, −3.70849048308428593812936501319, −2.79601580276702422997081623487, −0.857274569487811462988033166934, 0.04633864865943620649984461345, 2.38950210778554814943273164673, 3.08522870401622026305255721537, 3.91067262517487564503949347359, 5.06315324233338991971986389074, 5.94494659493628218483329183769, 6.88372289425049254791799305841, 7.52907526288249574834837865789, 8.273179917477335120012565048704, 9.064559249424299859355178949570

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