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Properties

Label 2-420e2-1.1-c1-0-298
Degree $2$
Conductor $176400$
Sign $-1$
Analytic cond. $1408.56$
Root an. cond. $37.5308$
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  + 3·11-s + 13-s − 6·17-s + 19-s − 9·23-s − 6·29-s − 8·31-s + 7·37-s − 3·41-s + 2·43-s − 9·47-s + 9·53-s + 8·61-s + 8·67-s + 4·73-s + 10·79-s − 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.904·11-s + 0.277·13-s − 1.45·17-s + 0.229·19-s − 1.87·23-s − 1.11·29-s − 1.43·31-s + 1.15·37-s − 0.468·41-s + 0.304·43-s − 1.31·47-s + 1.23·53-s + 1.02·61-s + 0.977·67-s + 0.468·73-s + 1.12·79-s − 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176400\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(1408.56\)
Root analytic conductor: \(37.5308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 176400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.51057839300334, −12.90609602147396, −12.57818536057741, −11.86913964221034, −11.53333696526462, −11.13624300893065, −10.69708877095277, −10.01414482779323, −9.533791111617936, −9.225132840848752, −8.661655988232838, −8.145687337327940, −7.720767510712703, −6.991752398247682, −6.684446810665085, −6.118641296414109, −5.645931016438718, −5.094511276597620, −4.339212381707551, −3.874648297937218, −3.660459160690009, −2.703936428500926, −1.974788136650491, −1.770309632627338, −0.7477704748886937, 0, 0.7477704748886937, 1.770309632627338, 1.974788136650491, 2.703936428500926, 3.660459160690009, 3.874648297937218, 4.339212381707551, 5.094511276597620, 5.645931016438718, 6.118641296414109, 6.684446810665085, 6.991752398247682, 7.720767510712703, 8.145687337327940, 8.661655988232838, 9.225132840848752, 9.533791111617936, 10.01414482779323, 10.69708877095277, 11.13624300893065, 11.53333696526462, 11.86913964221034, 12.57818536057741, 12.90609602147396, 13.51057839300334

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