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Properties

Label 2-12e3-72.61-c1-0-6
Degree $2$
Conductor $1728$
Sign $-0.250 - 0.968i$
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  + (1.5 − 0.866i)5-s + (−0.495 + 0.857i)7-s + (1.81 + 1.05i)11-s + (−5.50 + 3.18i)13-s − 3.81·17-s + 2i·19-s + (3.55 + 6.15i)23-s + (−1 + 1.73i)25-s + (−7.22 − 4.17i)29-s + (1.07 + 1.86i)31-s + 1.71i·35-s + 4.62i·37-s + (0.408 + 0.707i)41-s + (−1.97 − 1.14i)43-s + (−3.39 + 5.87i)47-s + ⋯
L(s)  = 1  + (0.670 − 0.387i)5-s + (−0.187 + 0.324i)7-s + (0.548 + 0.316i)11-s + (−1.52 + 0.882i)13-s − 0.925·17-s + 0.458i·19-s + (0.740 + 1.28i)23-s + (−0.200 + 0.346i)25-s + (−1.34 − 0.774i)29-s + (0.193 + 0.335i)31-s + 0.290i·35-s + 0.761i·37-s + (0.0637 + 0.110i)41-s + (−0.301 − 0.174i)43-s + (−0.494 + 0.857i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.250 - 0.968i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.250 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.186831704\)
\(L(\frac12)\) \(\approx\) \(1.186831704\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.495 - 0.857i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.81 - 1.05i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.50 - 3.18i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.81T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-3.55 - 6.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.22 + 4.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.07 - 1.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.62iT - 37T^{2} \)
41 \( 1 + (-0.408 - 0.707i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.97 + 1.14i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.39 - 5.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.14iT - 53T^{2} \)
59 \( 1 + (-10.3 + 5.95i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.22 + 2.43i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.8 + 6.86i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + (4.54 - 7.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.71 + 2.14i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 + (6.22 - 10.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.399406420646500154559657213839, −9.156595576467866877126034069509, −7.928612646923498810885398660994, −7.09805944291819050534994340909, −6.39184510141610661941738999162, −5.40497946358036510351488158427, −4.74372450904617224893842567622, −3.71310863150584388295826299347, −2.38525433393106527082874955831, −1.60260233865201003408845136152, 0.41685532457014746119841837690, 2.13397439139179969133139353717, 2.88603904314914369382778353551, 4.08485370374108857163560008786, 5.04529284813269288203693416956, 5.86384614885787728812244065509, 6.85974372039917076430149569967, 7.23043756365312111673319488008, 8.415958336593353233644204484629, 9.150372197686187014952219602391

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