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Properties

Label 2-12e3-144.59-c1-0-2
Degree $2$
Conductor $1728$
Sign $0.309 - 0.950i$
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  + (0.759 + 2.83i)5-s + (−1.41 − 2.45i)7-s + (0.212 − 0.794i)11-s + (−0.864 − 3.22i)13-s + 7.28i·17-s + (0.951 + 0.951i)19-s + (5.13 + 2.96i)23-s + (−3.12 + 1.80i)25-s + (0.473 − 1.76i)29-s + (2.05 + 1.18i)31-s + (5.87 − 5.87i)35-s + (6.03 + 6.03i)37-s + (−4.60 + 7.98i)41-s + (1.50 + 0.402i)43-s + (2.85 + 4.95i)47-s + ⋯
L(s)  = 1  + (0.339 + 1.26i)5-s + (−0.535 − 0.927i)7-s + (0.0641 − 0.239i)11-s + (−0.239 − 0.894i)13-s + 1.76i·17-s + (0.218 + 0.218i)19-s + (1.07 + 0.617i)23-s + (−0.624 + 0.360i)25-s + (0.0879 − 0.328i)29-s + (0.369 + 0.213i)31-s + (0.993 − 0.993i)35-s + (0.992 + 0.992i)37-s + (−0.719 + 1.24i)41-s + (0.229 + 0.0613i)43-s + (0.416 + 0.722i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.309 - 0.950i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.309 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.559583797\)
\(L(\frac12)\) \(\approx\) \(1.559583797\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.759 - 2.83i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.41 + 2.45i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.212 + 0.794i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.864 + 3.22i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 7.28iT - 17T^{2} \)
19 \( 1 + (-0.951 - 0.951i)T + 19iT^{2} \)
23 \( 1 + (-5.13 - 2.96i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.473 + 1.76i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-2.05 - 1.18i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.03 - 6.03i)T + 37iT^{2} \)
41 \( 1 + (4.60 - 7.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.50 - 0.402i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.85 - 4.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.50 + 4.50i)T - 53iT^{2} \)
59 \( 1 + (10.6 - 2.85i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-8.08 - 2.16i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (11.1 - 2.99i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.98iT - 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + (8.94 - 5.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.74 + 1.00i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 9.25T + 89T^{2} \)
97 \( 1 + (-0.148 - 0.257i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.937915878442684274582216374646, −8.593581025174383541671065017315, −7.77966999660273878578903833002, −7.02653233021179843318610737898, −6.34350648484528109708609886105, −5.67776314880306828121909584423, −4.34481793459801284772963066290, −3.37469005558764190122547744425, −2.78943063811346366622336341439, −1.24071913490170383409235928965, 0.64868925887787751124054239798, 2.07050322223930853574099921781, 2.98998216821770887939610496927, 4.44475056199666471565032720630, 5.03188087706387231386838958141, 5.77480345585952622805810648651, 6.78556639940244303211585789351, 7.52471032767182278733400974557, 8.812797019706036683997302662752, 9.120604724597402887695619623057

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