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Properties

Label 2-12e3-9.7-c1-0-17
Degree $2$
Conductor $1728$
Sign $-0.450 + 0.892i$
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  + (−1.18 − 2.05i)5-s + (2.18 − 3.78i)7-s + (0.5 − 0.866i)11-s + (−0.186 − 0.322i)13-s + 5.37·17-s + 0.627·19-s + (0.186 + 0.322i)23-s + (−0.313 + 0.543i)25-s + (2.18 − 3.78i)29-s + (3.18 + 5.51i)31-s − 10.3·35-s − 8.74·37-s + (−5.87 − 10.1i)41-s + (0.872 − 1.51i)43-s + (−2.18 + 3.78i)47-s + ⋯
L(s)  = 1  + (−0.530 − 0.918i)5-s + (0.826 − 1.43i)7-s + (0.150 − 0.261i)11-s + (−0.0516 − 0.0894i)13-s + 1.30·17-s + 0.144·19-s + (0.0388 + 0.0672i)23-s + (−0.0627 + 0.108i)25-s + (0.405 − 0.703i)29-s + (0.572 + 0.991i)31-s − 1.75·35-s − 1.43·37-s + (−0.917 − 1.58i)41-s + (0.133 − 0.230i)43-s + (−0.318 + 0.552i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.450 + 0.892i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.450 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.610705375\)
\(L(\frac12)\) \(\approx\) \(1.610705375\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 + 2.05i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.18 + 3.78i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.186 + 0.322i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.37T + 17T^{2} \)
19 \( 1 - 0.627T + 19T^{2} \)
23 \( 1 + (-0.186 - 0.322i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.18 + 3.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.18 - 5.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.74T + 37T^{2} \)
41 \( 1 + (5.87 + 10.1i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.872 + 1.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.18 - 3.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.744T + 53T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.18 + 2.05i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.87 - 3.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + (3.18 - 5.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.81 + 8.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (0.872 - 1.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.794198508666866610193400884084, −8.303856005265504874341945847115, −7.53669104690822631659440188552, −6.93340743970787468624586163192, −5.61558287306563807017637004879, −4.83493211487327447488618400362, −4.12672861421845498903483061222, −3.29083228647024529996166874760, −1.51796376350768886029042874634, −0.66204148524340279114253336402, 1.59947400193412848163765467526, 2.73358302946914140227990174579, 3.48688238497947763124654640055, 4.77454906781457827914668506398, 5.47914956472277432614705219304, 6.39225880601927591407317536053, 7.26172959146931970483179719512, 8.051570269311902644248130530496, 8.633466305236686895412742268168, 9.599540019339860452201176055935

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