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Properties

Label 2-420e2-1.1-c1-0-365
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  + 6·11-s − 5·13-s − 6·17-s + 5·19-s − 6·23-s + 6·29-s − 31-s + 2·37-s + 43-s − 6·47-s + 12·53-s + 6·59-s + 13·61-s − 11·67-s − 2·73-s − 8·79-s + 6·83-s + 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.80·11-s − 1.38·13-s − 1.45·17-s + 1.14·19-s − 1.25·23-s + 1.11·29-s − 0.179·31-s + 0.328·37-s + 0.152·43-s − 0.875·47-s + 1.64·53-s + 0.781·59-s + 1.66·61-s − 1.34·67-s − 0.234·73-s − 0.900·79-s + 0.658·83-s + 0.710·97-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 \( 1 - 6 T + p T^{2} \) 1.11.ag
13 \( 1 + 5 T + p T^{2} \) 1.13.f
17 \( 1 + 6 T + p T^{2} \) 1.17.g
19 \( 1 - 5 T + p T^{2} \) 1.19.af
23 \( 1 + 6 T + p T^{2} \) 1.23.g
29 \( 1 - 6 T + p T^{2} \) 1.29.ag
31 \( 1 + T + p T^{2} \) 1.31.b
37 \( 1 - 2 T + p T^{2} \) 1.37.ac
41 \( 1 + p T^{2} \) 1.41.a
43 \( 1 - T + p T^{2} \) 1.43.ab
47 \( 1 + 6 T + p T^{2} \) 1.47.g
53 \( 1 - 12 T + p T^{2} \) 1.53.am
59 \( 1 - 6 T + p T^{2} \) 1.59.ag
61 \( 1 - 13 T + p T^{2} \) 1.61.an
67 \( 1 + 11 T + p T^{2} \) 1.67.l
71 \( 1 + p T^{2} \) 1.71.a
73 \( 1 + 2 T + p T^{2} \) 1.73.c
79 \( 1 + 8 T + p T^{2} \) 1.79.i
83 \( 1 - 6 T + p T^{2} \) 1.83.ag
89 \( 1 + p T^{2} \) 1.89.a
97 \( 1 - 7 T + p T^{2} \) 1.97.ah
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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

−13.48476243863750, −12.98084793066856, −12.23208816792376, −12.02756598599667, −11.60401311088175, −11.27810095403304, −10.49948957771541, −9.944806259887740, −9.689012425767095, −9.184038344566689, −8.626483310118665, −8.310883696827969, −7.421006236987422, −7.194546699602097, −6.601425630884141, −6.256201334486955, −5.583569277222366, −4.976377030085651, −4.435695248005060, −4.020446631756189, −3.490660957276322, −2.634677736069731, −2.250351597602630, −1.513201414186824, −0.8460891068244995, 0, 0.8460891068244995, 1.513201414186824, 2.250351597602630, 2.634677736069731, 3.490660957276322, 4.020446631756189, 4.435695248005060, 4.976377030085651, 5.583569277222366, 6.256201334486955, 6.601425630884141, 7.194546699602097, 7.421006236987422, 8.310883696827969, 8.626483310118665, 9.184038344566689, 9.689012425767095, 9.944806259887740, 10.49948957771541, 11.27810095403304, 11.60401311088175, 12.02756598599667, 12.23208816792376, 12.98084793066856, 13.48476243863750

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