Properties

Label 2-12e3-8.5-c3-0-45
Degree $2$
Conductor $1728$
Sign $0.707 - 0.707i$
Analytic cond. $101.955$
Root an. cond. $10.0972$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.3i·5-s + 21.4·7-s − 24.7i·11-s + 21.4i·13-s − 123.·17-s + 7i·19-s + 114.·23-s + 17·25-s − 270. i·29-s + 214.·31-s + 222. i·35-s + 235. i·37-s + 395.·41-s + 92i·43-s + 114.·47-s + ⋯
L(s)  = 1  + 0.929i·5-s + 1.15·7-s − 0.678i·11-s + 0.457i·13-s − 1.76·17-s + 0.0845i·19-s + 1.03·23-s + 0.136·25-s − 1.73i·29-s + 1.24·31-s + 1.07i·35-s + 1.04i·37-s + 1.50·41-s + 0.326i·43-s + 0.354·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(101.955\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :3/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.498416972\)
\(L(\frac12)\) \(\approx\) \(2.498416972\)
\(L(\frac{5}{2})\) 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 \( 1 - 10.3iT - 125T^{2} \)
7 \( 1 - 21.4T + 343T^{2} \)
11 \( 1 + 24.7iT - 1.33e3T^{2} \)
13 \( 1 - 21.4iT - 2.19e3T^{2} \)
17 \( 1 + 123.T + 4.91e3T^{2} \)
19 \( 1 - 7iT - 6.85e3T^{2} \)
23 \( 1 - 114.T + 1.21e4T^{2} \)
29 \( 1 + 270. iT - 2.43e4T^{2} \)
31 \( 1 - 214.T + 2.97e4T^{2} \)
37 \( 1 - 235. iT - 5.06e4T^{2} \)
41 \( 1 - 395.T + 6.89e4T^{2} \)
43 \( 1 - 92iT - 7.95e4T^{2} \)
47 \( 1 - 114.T + 1.03e5T^{2} \)
53 \( 1 - 20.7iT - 1.48e5T^{2} \)
59 \( 1 + 173. iT - 2.05e5T^{2} \)
61 \( 1 - 449. iT - 2.26e5T^{2} \)
67 \( 1 - 353iT - 3.00e5T^{2} \)
71 \( 1 - 789.T + 3.57e5T^{2} \)
73 \( 1 - 425T + 3.89e5T^{2} \)
79 \( 1 + 1.30e3T + 4.93e5T^{2} \)
83 \( 1 + 593. iT - 5.71e5T^{2} \)
89 \( 1 + 74.2T + 7.04e5T^{2} \)
97 \( 1 - 799T + 9.12e5T^{2} \)
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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

−8.893308913300068972871368284645, −8.320502297013802752926618627209, −7.45653545535827841104224983210, −6.63654481245709755838997468587, −6.01494382158066661575157910923, −4.77755897586499018372852540673, −4.21756551824701942466429975208, −2.89867117372510394655084949261, −2.18026715244759480572893873080, −0.872086490516631685917743095055, 0.68730973755729212158423443148, 1.65314388110765632780005883399, 2.64583575798987003867496119028, 4.13031814483639566291610030071, 4.83380984294581961842570635945, 5.26318590732796282129803857638, 6.55145820981181313171295836003, 7.33967006668188232667589720820, 8.198978152416400836843307653384, 8.875467405557860802291557816465

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