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Properties

Label 2-1872-13.10-c1-0-11
Degree $2$
Conductor $1872$
Sign $0.964 - 0.265i$
Analytic cond. $14.9479$
Root an. cond. $3.86626$
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  − 1.73i·5-s + (−2.5 + 2.59i)13-s + (−1.5 + 2.59i)17-s + (3 + 1.73i)19-s + (3 + 5.19i)23-s + 2.00·25-s + (1.5 + 2.59i)29-s + 3.46i·31-s + (7.5 − 4.33i)37-s + (4.5 − 2.59i)41-s + (4 − 6.92i)43-s + 3.46i·47-s + (−3.5 − 6.06i)49-s + 3·53-s + (6 + 3.46i)59-s + ⋯
L(s)  = 1  − 0.774i·5-s + (−0.693 + 0.720i)13-s + (−0.363 + 0.630i)17-s + (0.688 + 0.397i)19-s + (0.625 + 1.08i)23-s + 0.400·25-s + (0.278 + 0.482i)29-s + 0.622i·31-s + (1.23 − 0.711i)37-s + (0.702 − 0.405i)41-s + (0.609 − 1.05i)43-s + 0.505i·47-s + (−0.5 − 0.866i)49-s + 0.412·53-s + (0.781 + 0.450i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(14.9479\)
Root analytic conductor: \(3.86626\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.649925025\)
\(L(\frac12)\) \(\approx\) \(1.649925025\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (2.5 - 2.59i)T \)
good5 \( 1 + 1.73iT - 5T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (-7.5 + 4.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-6 - 3.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6 - 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.123077605873459863605724253906, −8.697988548714246519582033478004, −7.61857570742759708878695674415, −7.04737512891881071243488383493, −5.96159616064971977788446866911, −5.16666675086398900658768968531, −4.41897364129475346952967322849, −3.45064800798618838084577413872, −2.18501010135135955767249891651, −1.04391732407829540658714043117, 0.75508942227275846024388682668, 2.58162833387306437816196125022, 2.94410895101607160802272327130, 4.33594373541198768671724370219, 5.07917652869731294007720257944, 6.12922487867576987838812897419, 6.85481181379165610606297371129, 7.57821239145530444057098171169, 8.299138643179058367045747390571, 9.393889983743260479636102295743

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