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Properties

Label 2-2592-36.23-c1-0-25
Degree $2$
Conductor $2592$
Sign $0.984 - 0.173i$
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  + (3.47 + 2.00i)5-s + (2.66 − 1.54i)7-s + (−2.97 − 5.15i)11-s + (0.673 − 1.16i)13-s + 4.87i·17-s + 7.49i·19-s + (1.28 − 2.23i)23-s + (5.57 + 9.65i)25-s + (5.60 − 3.23i)29-s + (−1.28 − 0.743i)31-s + 12.3·35-s + 1.91·37-s + (−1.69 − 0.978i)41-s + (6.79 − 3.92i)43-s + (1.14 + 1.97i)47-s + ⋯
L(s)  = 1  + (1.55 + 0.898i)5-s + (1.00 − 0.582i)7-s + (−0.897 − 1.55i)11-s + (0.186 − 0.323i)13-s + 1.18i·17-s + 1.71i·19-s + (0.268 − 0.465i)23-s + (1.11 + 1.93i)25-s + (1.04 − 0.600i)29-s + (−0.231 − 0.133i)31-s + 2.09·35-s + 0.315·37-s + (−0.264 − 0.152i)41-s + (1.03 − 0.598i)43-s + (0.166 + 0.288i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.770476113\)
\(L(\frac12)\) \(\approx\) \(2.770476113\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.47 - 2.00i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.66 + 1.54i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.97 + 5.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.673 + 1.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.87iT - 17T^{2} \)
19 \( 1 - 7.49iT - 19T^{2} \)
23 \( 1 + (-1.28 + 2.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.60 + 3.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.28 + 0.743i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.91T + 37T^{2} \)
41 \( 1 + (1.69 + 0.978i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.79 + 3.92i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.14 - 1.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.871iT - 53T^{2} \)
59 \( 1 + (-0.650 + 1.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.29 - 5.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.44 - 4.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.64T + 71T^{2} \)
73 \( 1 + 2.14T + 73T^{2} \)
79 \( 1 + (-8.08 + 4.66i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.09 + 8.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.947iT - 89T^{2} \)
97 \( 1 + (5.15 + 8.93i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.731320988289535918566784919697, −8.183076796415592722928778985558, −7.47555450522355243940387980816, −6.28710878681731137460553417102, −5.91668651612375319273872713909, −5.27873887561089373554989174143, −4.01174508546998875413823340621, −3.05600867669114899154856781891, −2.14293528693536373319296546910, −1.15532499253498148061827781296, 1.12017346843600912816695238848, 2.16781810059198202866422807815, 2.61554687001850437465123287087, 4.64523029379681152817901236265, 4.95015399121056364998657831060, 5.42121507826044351858444775455, 6.57584538604306492224265071084, 7.29078069238096402890211867913, 8.244291868700137073191680521889, 9.085381409867922853301874569718

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