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Properties

Label 2-120e2-1.1-c1-0-42
Degree $2$
Conductor $14400$
Sign $1$
Analytic cond. $114.984$
Root an. cond. $10.7230$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·7-s − 5·13-s + 19-s − 7·31-s + 10·37-s − 5·43-s + 18·49-s + 13·61-s − 5·67-s − 10·73-s − 4·79-s − 25·91-s + 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.88·7-s − 1.38·13-s + 0.229·19-s − 1.25·31-s + 1.64·37-s − 0.762·43-s + 18/7·49-s + 1.66·61-s − 0.610·67-s − 1.17·73-s − 0.450·79-s − 2.62·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(114.984\)
Root analytic conductor: \(10.7230\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 14400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.563326940\)
\(L(\frac12)\) \(\approx\) \(2.563326940\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−16.17299226480994, −15.37006410013681, −14.83890559416255, −14.46672726165482, −14.19312822316627, −13.24974319430971, −12.80227731513365, −11.95500382856174, −11.62192911864154, −11.13737100980921, −10.46044457285436, −9.864854431330399, −9.184811760773533, −8.526228950680148, −7.929378649379829, −7.437992287266115, −6.967797640581750, −5.840408005738872, −5.342789801038731, −4.657231614647301, −4.298021066552855, −3.208986402721924, −2.259693877568981, −1.764741436286940, −0.7099203982221574, 0.7099203982221574, 1.764741436286940, 2.259693877568981, 3.208986402721924, 4.298021066552855, 4.657231614647301, 5.342789801038731, 5.840408005738872, 6.967797640581750, 7.437992287266115, 7.929378649379829, 8.526228950680148, 9.184811760773533, 9.864854431330399, 10.46044457285436, 11.13737100980921, 11.62192911864154, 11.95500382856174, 12.80227731513365, 13.24974319430971, 14.19312822316627, 14.46672726165482, 14.83890559416255, 15.37006410013681, 16.17299226480994

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