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Properties

Label 2-12e3-144.133-c1-0-1
Degree $2$
Conductor $1728$
Sign $-0.442 - 0.896i$
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  + (0.326 − 1.21i)5-s + (−0.707 − 0.408i)7-s + (−1.85 + 0.497i)11-s + (−0.434 − 0.116i)13-s − 6.62·17-s + (−1.18 + 1.18i)19-s + (−2.66 + 1.53i)23-s + (2.95 + 1.70i)25-s + (2.31 + 8.65i)29-s + (4.61 + 7.99i)31-s + (−0.728 + 0.728i)35-s + (−2.14 − 2.14i)37-s + (−9.15 + 5.28i)41-s + (6.19 − 1.66i)43-s + (−0.140 + 0.244i)47-s + ⋯
L(s)  = 1  + (0.145 − 0.544i)5-s + (−0.267 − 0.154i)7-s + (−0.559 + 0.149i)11-s + (−0.120 − 0.0322i)13-s − 1.60·17-s + (−0.271 + 0.271i)19-s + (−0.555 + 0.320i)23-s + (0.591 + 0.341i)25-s + (0.430 + 1.60i)29-s + (0.828 + 1.43i)31-s + (−0.123 + 0.123i)35-s + (−0.352 − 0.352i)37-s + (−1.43 + 0.825i)41-s + (0.945 − 0.253i)43-s + (−0.0205 + 0.0356i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.442 - 0.896i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1585, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.442 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6365902723\)
\(L(\frac12)\) \(\approx\) \(0.6365902723\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.326 + 1.21i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.707 + 0.408i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.85 - 0.497i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.434 + 0.116i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 6.62T + 17T^{2} \)
19 \( 1 + (1.18 - 1.18i)T - 19iT^{2} \)
23 \( 1 + (2.66 - 1.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.31 - 8.65i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-4.61 - 7.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.14 + 2.14i)T + 37iT^{2} \)
41 \( 1 + (9.15 - 5.28i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.19 + 1.66i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.140 - 0.244i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.83 + 4.83i)T + 53iT^{2} \)
59 \( 1 + (-1.91 + 7.15i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.64 - 9.87i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-5.22 - 1.39i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.27iT - 71T^{2} \)
73 \( 1 + 4.92iT - 73T^{2} \)
79 \( 1 + (7.70 - 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.92 - 10.9i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 3.44iT - 89T^{2} \)
97 \( 1 + (4.46 - 7.74i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.533647258426077677963624032863, −8.679088265972085305672350409901, −8.249905690546889347087740395640, −6.98029727477855991574881831820, −6.59220285953420543091725060050, −5.29781834154791971018840423136, −4.80727561446527482921366198146, −3.71104949147042527495819407506, −2.61756963243927827001672903262, −1.43355763831914859765671827147, 0.23009505842945279449076020509, 2.19938349244715190230794814186, 2.79898935364084765108344987179, 4.11597717483190858181399405432, 4.84163309378403756075992452112, 6.14622097175472398179349067333, 6.44961792104896362930547339598, 7.48860060777028862924387660627, 8.298789743417421399824027674623, 9.046540222672458618449032391337

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