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Properties

Label 2-2592-36.11-c1-0-27
Degree $2$
Conductor $2592$
Sign $-0.422 + 0.906i$
Analytic cond. $20.6972$
Root an. cond. $4.54942$
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  + (−2.72 + 1.57i)5-s + (−2.98 − 1.72i)7-s + (2.28 − 3.94i)11-s + (3.44 + 5.97i)13-s + 3.46i·17-s + 4.89i·19-s + (−1.41 − 2.44i)23-s + (2.44 − 4.24i)25-s + (−1.89 − 1.09i)29-s + (−2.20 + 1.27i)31-s + 10.8·35-s − 4.89·37-s + (3.55 − 2.04i)41-s + (−2.51 − 1.44i)43-s + (1.09 − 1.89i)47-s + ⋯
L(s)  = 1  + (−1.21 + 0.703i)5-s + (−1.12 − 0.651i)7-s + (0.687 − 1.19i)11-s + (0.956 + 1.65i)13-s + 0.840i·17-s + 1.12i·19-s + (−0.294 − 0.510i)23-s + (0.489 − 0.848i)25-s + (−0.352 − 0.203i)29-s + (−0.396 + 0.229i)31-s + 1.83·35-s − 0.805·37-s + (0.554 − 0.320i)41-s + (−0.382 − 0.221i)43-s + (0.159 − 0.276i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.422 + 0.906i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.422 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3855801022\)
\(L(\frac12)\) \(\approx\) \(0.3855801022\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.72 - 1.57i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.98 + 1.72i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.28 + 3.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.44 - 5.97i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 - 4.89iT - 19T^{2} \)
23 \( 1 + (1.41 + 2.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.89 + 1.09i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.20 - 1.27i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.89T + 37T^{2} \)
41 \( 1 + (-3.55 + 2.04i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.51 + 1.44i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.09 + 1.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 12.9iT - 53T^{2} \)
59 \( 1 + (1.09 + 1.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.9 + 7.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 7.89T + 73T^{2} \)
79 \( 1 + (1.73 + i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.20 + 10.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.02iT - 89T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.603546486472801935204722080056, −7.906195743351357877173268429692, −6.88658491642479163611596826065, −6.54609019531271851974644172075, −5.83540450952158586787331849621, −4.15667729791278893244788768221, −3.74005603720894620451458248681, −3.32900720338297893904118484969, −1.66737566501829165651344077054, −0.15300350600767965497788658342, 1.04626305702053012275880647841, 2.69688516408654417066593575662, 3.48769143481958054455935530069, 4.29801007749918711907874987738, 5.19547722296994399799959554104, 5.99672604352625575322146561734, 6.98683825555152623678136573449, 7.58227796230376582892533278038, 8.410165906610239079305945276642, 9.156603202041477741926586899639

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