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Properties

Label 2-2064-172.123-c1-0-29
Degree $2$
Conductor $2064$
Sign $-0.216 + 0.976i$
Analytic cond. $16.4811$
Root an. cond. $4.05969$
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  + (0.5 − 0.866i)3-s + (−1 − 1.73i)7-s + (−0.499 − 0.866i)9-s − 1.73i·11-s + (2 + 3.46i)13-s + (1.5 + 2.59i)17-s + (2.5 − 4.33i)19-s − 1.99·21-s + (−2.5 − 4.33i)25-s − 0.999·27-s + (6 − 3.46i)29-s + (6 + 3.46i)31-s + (−1.49 − 0.866i)33-s + (−9 − 5.19i)37-s + 3.99·39-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.377 − 0.654i)7-s + (−0.166 − 0.288i)9-s − 0.522i·11-s + (0.554 + 0.960i)13-s + (0.363 + 0.630i)17-s + (0.573 − 0.993i)19-s − 0.436·21-s + (−0.5 − 0.866i)25-s − 0.192·27-s + (1.11 − 0.643i)29-s + (1.07 + 0.622i)31-s + (−0.261 − 0.150i)33-s + (−1.47 − 0.854i)37-s + 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2064\)    =    \(2^{4} \cdot 3 \cdot 43\)
Sign: $-0.216 + 0.976i$
Analytic conductor: \(16.4811\)
Root analytic conductor: \(4.05969\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2064} (1327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2064,\ (\ :1/2),\ -0.216 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.675274445\)
\(L(\frac12)\) \(\approx\) \(1.675274445\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (6.5 + 0.866i)T \)
good5 \( 1 + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.73iT - 11T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6 + 3.46i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6 - 3.46i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (9 + 5.19i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12.1iT - 59T^{2} \)
61 \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 + 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9 - 5.19i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (13.5 + 7.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−8.678416042513499212558775599768, −8.288548559623850645897907655349, −7.21208853856482387279566237943, −6.64868953894255843574287275359, −5.96249525075114631093420376686, −4.77099543724488898803866562340, −3.83740960162663643351966380841, −3.03429767796366034583139065541, −1.81840704266177335983209626701, −0.60504367291015725885429090425, 1.37559881505653221777780712643, 2.82714852429673779044783661557, 3.34655814602942127903908275633, 4.50285616855673463150247714963, 5.39476300153005676641177879243, 6.01037265062432050219270190009, 7.07042159928438587183662380047, 7.963995568884317381031865370552, 8.550591770001251899035978975604, 9.448377545504892029089703955271

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