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Properties

Label 2-10800-1.1-c1-0-48
Degree $2$
Conductor $10800$
Sign $-1$
Analytic cond. $86.2384$
Root an. cond. $9.28646$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 5·11-s − 3·13-s + 17-s + 6·19-s − 23-s + 9·29-s + 5·31-s − 2·37-s + 2·41-s − 43-s + 13·47-s + 9·49-s − 4·59-s + 8·61-s + 4·67-s − 6·71-s − 2·73-s + 20·77-s − 9·79-s − 4·83-s + 14·89-s + 12·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.50·11-s − 0.832·13-s + 0.242·17-s + 1.37·19-s − 0.208·23-s + 1.67·29-s + 0.898·31-s − 0.328·37-s + 0.312·41-s − 0.152·43-s + 1.89·47-s + 9/7·49-s − 0.520·59-s + 1.02·61-s + 0.488·67-s − 0.712·71-s − 0.234·73-s + 2.27·77-s − 1.01·79-s − 0.439·83-s + 1.48·89-s + 1.25·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10800\)    =    \(2^{4} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(86.2384\)
Root analytic conductor: \(9.28646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10800,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−16.67109887925480, −16.05740816298920, −15.77241800026784, −15.40234164610917, −14.44453829693353, −13.83074316567614, −13.40021064578348, −12.80138573138099, −12.17534837278418, −11.91124335230121, −10.83201404957183, −10.17864295883731, −9.964380492552931, −9.332157140823814, −8.531430861732152, −7.806345288122489, −7.256342092045362, −6.651099201454534, −5.851690632205178, −5.302168510663194, −4.585910020975415, −3.616768208862752, −2.709419563979706, −2.649937082855098, −0.9792056128982597, 0, 0.9792056128982597, 2.649937082855098, 2.709419563979706, 3.616768208862752, 4.585910020975415, 5.302168510663194, 5.851690632205178, 6.651099201454534, 7.256342092045362, 7.806345288122489, 8.531430861732152, 9.332157140823814, 9.964380492552931, 10.17864295883731, 10.83201404957183, 11.91124335230121, 12.17534837278418, 12.80138573138099, 13.40021064578348, 13.83074316567614, 14.44453829693353, 15.40234164610917, 15.77241800026784, 16.05740816298920, 16.67109887925480

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