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Properties

Label 2-12e3-144.133-c1-0-6
Degree $2$
Conductor $1728$
Sign $0.0436 - 0.999i$
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 + 3.73i)5-s + (−0.633 − 0.366i)7-s + (2.86 − 0.767i)11-s + (6.09 + 1.63i)13-s + 2.26·17-s + (0.633 − 0.633i)19-s + (1.09 − 0.633i)23-s + (−8.59 − 4.96i)25-s + (−0.633 − 2.36i)29-s + (3.73 + 6.46i)31-s + (2 − 2i)35-s + (1.26 + 1.26i)37-s + (−2.59 + 1.5i)41-s + (1.23 − 0.330i)43-s + (−4.83 + 8.36i)47-s + ⋯
L(s)  = 1  + (−0.447 + 1.66i)5-s + (−0.239 − 0.138i)7-s + (0.864 − 0.231i)11-s + (1.69 + 0.453i)13-s + 0.550·17-s + (0.145 − 0.145i)19-s + (0.228 − 0.132i)23-s + (−1.71 − 0.992i)25-s + (−0.117 − 0.439i)29-s + (0.670 + 1.16i)31-s + (0.338 − 0.338i)35-s + (0.208 + 0.208i)37-s + (−0.405 + 0.234i)41-s + (0.187 − 0.0503i)43-s + (−0.704 + 1.22i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0436 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0436 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.0436 - 0.999i$
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: \((2,\ 1728,\ (\ :1/2),\ 0.0436 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.694137891\)
\(L(\frac12)\) \(\approx\) \(1.694137891\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 3.73i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.633 + 0.366i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.86 + 0.767i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-6.09 - 1.63i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 2.26T + 17T^{2} \)
19 \( 1 + (-0.633 + 0.633i)T - 19iT^{2} \)
23 \( 1 + (-1.09 + 0.633i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.633 + 2.36i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-3.73 - 6.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 - 1.26i)T + 37iT^{2} \)
41 \( 1 + (2.59 - 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.23 + 0.330i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (4.83 - 8.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.535 - 0.535i)T + 53iT^{2} \)
59 \( 1 + (1.33 - 4.96i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.803 - 3i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-5.23 - 1.40i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 - 9.73iT - 73T^{2} \)
79 \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.366 - 1.36i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 + (4.13 - 7.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.620756770168070227199172271429, −8.634299490260216176265056361699, −7.922496211004158647687211911843, −6.80449250273389071687045354783, −6.61066232324852475097844279835, −5.73878116097684680097118464735, −4.21977933441408093319507163962, −3.52233089182616022207573523700, −2.85272185069481697134385075933, −1.31933835114163033168861052078, 0.75824261685559736102049353203, 1.62724581939789917924057996070, 3.43193989464169651925909760574, 4.06305278238611565285353246796, 5.02388676061866932290372354591, 5.80621193273658517338743310043, 6.61566987415076685668379069099, 7.911204701159158097871661665197, 8.298059296746447792189479829794, 9.152761941898776517771323275547

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