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Properties

Label 2-12e3-24.11-c1-0-8
Degree $2$
Conductor $1728$
Sign $-0.258 - 0.965i$
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  + 5i·7-s − 1.73i·13-s + 8.66·19-s − 5·25-s + 4i·31-s + 12.1i·37-s − 10.3·43-s − 18·49-s + 8.66i·61-s + 12.1·67-s − 17·73-s + 13i·79-s + 8.66·91-s + 5·97-s + 7i·103-s + ⋯
L(s)  = 1  + 1.88i·7-s − 0.480i·13-s + 1.98·19-s − 25-s + 0.718i·31-s + 1.99i·37-s − 1.58·43-s − 2.57·49-s + 1.10i·61-s + 1.48·67-s − 1.98·73-s + 1.46i·79-s + 0.907·91-s + 0.507·97-s + 0.689i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.258 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.439505376\)
\(L(\frac12)\) \(\approx\) \(1.439505376\)
\(L(\frac{3}{2})\) 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 \( 1 + 5T^{2} \)
7 \( 1 - 5iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 8.66T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 - 12.1iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 8.66iT - 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 17T + 73T^{2} \)
79 \( 1 - 13iT - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 5T + 97T^{2} \)
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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

−9.656916904137509506442920125660, −8.652047287583499481959448251100, −8.194496494539749818274406462836, −7.19352524995462728741058887431, −6.17204030992571749496696385367, −5.46318648807501956774462476444, −4.91769755750362118283623603314, −3.37161987370348695422495352831, −2.72199205244450967238288825978, −1.50507068338630929629015783506, 0.55934927073716969417253661694, 1.74974904866685135080557043354, 3.33651648721280338232712818189, 3.96982901749424590681987402081, 4.86728643190587743460009391774, 5.87312602209894609882244381442, 6.94734096121274654913570190210, 7.43804884542546204264318514351, 8.061746391442562847853142554632, 9.332642900199456248268642040452

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