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Properties

Label 2-2160-5.4-c1-0-36
Degree $2$
Conductor $2160$
Sign $-0.223 + 0.974i$
Analytic cond. $17.2476$
Root an. cond. $4.15303$
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  + (−2.17 − 0.5i)5-s − 4.35i·7-s + 4.35·11-s + 4i·17-s + 6·19-s − 2i·23-s + (4.50 + 2.17i)25-s − 7·31-s + (−2.17 + 9.50i)35-s − 8.71i·37-s + 8.71·41-s − 8.71i·43-s − 2i·47-s − 12.0·49-s + 3i·53-s + ⋯
L(s)  = 1  + (−0.974 − 0.223i)5-s − 1.64i·7-s + 1.31·11-s + 0.970i·17-s + 1.37·19-s − 0.417i·23-s + (0.900 + 0.435i)25-s − 1.25·31-s + (−0.368 + 1.60i)35-s − 1.43i·37-s + 1.36·41-s − 1.32i·43-s − 0.291i·47-s − 1.71·49-s + 0.412i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.223 + 0.974i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ -0.223 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.379083045\)
\(L(\frac12)\) \(\approx\) \(1.379083045\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2.17 + 0.5i)T \)
good7 \( 1 + 4.35iT - 7T^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + 8.71iT - 37T^{2} \)
41 \( 1 - 8.71T + 41T^{2} \)
43 \( 1 + 8.71iT - 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 - 3iT - 53T^{2} \)
59 \( 1 + 8.71T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 8.71iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 4.35iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 5iT - 83T^{2} \)
89 \( 1 + 8.71T + 89T^{2} \)
97 \( 1 + 4.35iT - 97T^{2} \)
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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

−8.930483787376776350702799234112, −7.84449051851657256034969005688, −7.38195793091336561977050901603, −6.76210546775078375208825243681, −5.70442166840490137390538396995, −4.49990142408601185397172809092, −3.91419685220581800678477012977, −3.40413890904809460604931686374, −1.54061628817866170653356116432, −0.55894289828552607051445944650, 1.29185300653474010907580010010, 2.72344532496388764028809155494, 3.37656389073463306410437775450, 4.46896452847844228530078010632, 5.33965992042105433312300028155, 6.14760465331538496449936466691, 7.02623194236618650724285487525, 7.75150713100437454696101103661, 8.592433273759511904899004132625, 9.323522482786687459505638901685

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