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Properties

Label 2-420e2-1.1-c1-0-301
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  − 13-s − 3·17-s + 2·19-s − 3·23-s − 3·29-s − 31-s − 2·37-s − 3·41-s − 7·43-s + 6·47-s + 9·53-s − 3·59-s + 61-s + 8·67-s − 4·73-s − 8·79-s + 15·83-s + 6·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.277·13-s − 0.727·17-s + 0.458·19-s − 0.625·23-s − 0.557·29-s − 0.179·31-s − 0.328·37-s − 0.468·41-s − 1.06·43-s + 0.875·47-s + 1.23·53-s − 0.390·59-s + 0.128·61-s + 0.977·67-s − 0.468·73-s − 0.900·79-s + 1.64·83-s + 0.635·89-s + 0.812·97-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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \) 1.11.a
13 \( 1 + T + p T^{2} \) 1.13.b
17 \( 1 + 3 T + p T^{2} \) 1.17.d
19 \( 1 - 2 T + p T^{2} \) 1.19.ac
23 \( 1 + 3 T + p T^{2} \) 1.23.d
29 \( 1 + 3 T + p T^{2} \) 1.29.d
31 \( 1 + T + p T^{2} \) 1.31.b
37 \( 1 + 2 T + p T^{2} \) 1.37.c
41 \( 1 + 3 T + p T^{2} \) 1.41.d
43 \( 1 + 7 T + p T^{2} \) 1.43.h
47 \( 1 - 6 T + p T^{2} \) 1.47.ag
53 \( 1 - 9 T + p T^{2} \) 1.53.aj
59 \( 1 + 3 T + p T^{2} \) 1.59.d
61 \( 1 - T + p T^{2} \) 1.61.ab
67 \( 1 - 8 T + p T^{2} \) 1.67.ai
71 \( 1 + p T^{2} \) 1.71.a
73 \( 1 + 4 T + p T^{2} \) 1.73.e
79 \( 1 + 8 T + p T^{2} \) 1.79.i
83 \( 1 - 15 T + p T^{2} \) 1.83.ap
89 \( 1 - 6 T + p T^{2} \) 1.89.ag
97 \( 1 - 8 T + p T^{2} \) 1.97.ai
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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

−13.50021461159209, −13.03906642466091, −12.36930719203186, −12.02415403422624, −11.59761253838464, −11.02905380939939, −10.64688294416134, −9.941293019490492, −9.815331984913899, −8.978558122238730, −8.779757298207559, −8.143638493794041, −7.595760655875791, −7.166561194767019, −6.651356596744033, −6.125348233496245, −5.540114235711455, −5.072533852574598, −4.536817872798309, −3.867116924594781, −3.492846482030478, −2.721955591237241, −2.153064712534924, −1.626192808682146, −0.7491736046514964, 0, 0.7491736046514964, 1.626192808682146, 2.153064712534924, 2.721955591237241, 3.492846482030478, 3.867116924594781, 4.536817872798309, 5.072533852574598, 5.540114235711455, 6.125348233496245, 6.651356596744033, 7.166561194767019, 7.595760655875791, 8.143638493794041, 8.779757298207559, 8.978558122238730, 9.815331984913899, 9.941293019490492, 10.64688294416134, 11.02905380939939, 11.59761253838464, 12.02415403422624, 12.36930719203186, 13.03906642466091, 13.50021461159209

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