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Properties

Label 2-12e3-24.5-c2-0-12
Degree $2$
Conductor $1728$
Sign $-0.707 - 0.707i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.19·7-s + 25.9i·13-s + 11i·19-s − 25·25-s − 41.5·31-s − 57.1i·37-s + 22i·43-s − 22·49-s + 15.5i·61-s + 109i·67-s − 97·73-s + 88.3·79-s + 135i·91-s − 167·97-s + 202.·103-s + ⋯
L(s)  = 1  + 0.742·7-s + 1.99i·13-s + 0.578i·19-s − 25-s − 1.34·31-s − 1.54i·37-s + 0.511i·43-s − 0.448·49-s + 0.255i·61-s + 1.62i·67-s − 1.32·73-s + 1.11·79-s + 1.48i·91-s − 1.72·97-s + 1.96·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.140970326\)
\(L(\frac12)\) \(\approx\) \(1.140970326\)
\(L(2)\) 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 \( 1 + 25T^{2} \)
7 \( 1 - 5.19T + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 - 25.9iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 11iT - 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + 41.5T + 961T^{2} \)
37 \( 1 + 57.1iT - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 22iT - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + 3.48e3T^{2} \)
61 \( 1 - 15.5iT - 3.72e3T^{2} \)
67 \( 1 - 109iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 97T + 5.32e3T^{2} \)
79 \( 1 - 88.3T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 167T + 9.40e3T^{2} \)
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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

−9.300002306884565664090727249061, −8.769508919905099144569779785642, −7.77549677790242400551839218389, −7.14841568885234731206681963356, −6.21958724293408002342594003677, −5.36897378024111961227105300580, −4.35709884573216801902004522005, −3.76783470059572570566200213757, −2.22282462162336382155923245887, −1.51387794702189989953977249665, 0.28867666947777080170707471485, 1.57977895742488037898445304379, 2.79634584167771325621804483771, 3.69657950857931745068457954687, 4.89109681489395960392544966517, 5.45410050542516433914827357127, 6.36316732501943655243870964109, 7.50907917763943441939933680610, 7.962370469527106242750504851037, 8.723668282434916355006147020050

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