↑

Properties

Label 2-3744-13.12-c1-0-23
Degree $2$
Conductor $3744$
Sign $-0.246 - 0.969i$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.49i·5-s + 1.10i·7-s + 3.60i·11-s + (3.49 − 0.890i)13-s − 2·17-s − 1.10i·19-s + 7.20·23-s − 1.21·25-s + 5.20·29-s + 6.09i·31-s − 2.76·35-s − 11.2i·37-s + 7.48i·41-s + 9.42·43-s + 7.60i·47-s + ⋯
L(s)  = 1  + 1.11i·5-s + 0.419i·7-s + 1.08i·11-s + (0.969 − 0.246i)13-s − 0.485·17-s − 0.254i·19-s + 1.50·23-s − 0.243·25-s + 0.967·29-s + 1.09i·31-s − 0.467·35-s − 1.84i·37-s + 1.16i·41-s + 1.43·43-s + 1.10i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.246 - 0.969i$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ -0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.939898487\)
\(L(\frac12)\) \(\approx\) \(1.939898487\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.49 + 0.890i)T \)
good5 \( 1 - 2.49iT - 5T^{2} \)
7 \( 1 - 1.10iT - 7T^{2} \)
11 \( 1 - 3.60iT - 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 1.10iT - 19T^{2} \)
23 \( 1 - 7.20T + 23T^{2} \)
29 \( 1 - 5.20T + 29T^{2} \)
31 \( 1 - 6.09iT - 31T^{2} \)
37 \( 1 + 11.2iT - 37T^{2} \)
41 \( 1 - 7.48iT - 41T^{2} \)
43 \( 1 - 9.42T + 43T^{2} \)
47 \( 1 - 7.60iT - 47T^{2} \)
53 \( 1 + 4.76T + 53T^{2} \)
59 \( 1 + 1.38iT - 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 - 11.8iT - 67T^{2} \)
71 \( 1 + 1.82iT - 71T^{2} \)
73 \( 1 + 4.98iT - 73T^{2} \)
79 \( 1 - 0.439T + 79T^{2} \)
83 \( 1 - 0.591iT - 83T^{2} \)
89 \( 1 - 1.06iT - 89T^{2} \)
97 \( 1 - 3.01iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.898069589367349714658212400430, −7.86564338747975383808814641024, −7.13389473539864331019440202579, −6.62756818517701551331759390961, −5.88062046973120450543327046320, −4.92322587568926346984298351794, −4.12390833041535023152819713224, −3.03074869785478444462703760658, −2.52988889555138227988619230165, −1.26412455998469814943594960395, 0.65778630066444938655397007697, 1.36837807882616828669429223473, 2.78526834288783365915698990580, 3.73874962781513430449925281904, 4.49252696822193868730798132719, 5.24483343868409091685548579144, 6.05823102151408994116330772136, 6.73165033135275169603996496871, 7.70742148866971143644795806267, 8.542861971290426997483352716639

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