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Properties

Label 2-168e2-1.1-c1-0-132
Degree $2$
Conductor $28224$
Sign $-1$
Analytic cond. $225.369$
Root an. cond. $15.0123$
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  + 2·5-s − 4·13-s + 2·17-s − 25-s + 4·29-s − 2·37-s − 10·41-s − 4·53-s + 12·61-s − 8·65-s + 16·73-s + 4·85-s − 10·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.10·13-s + 0.485·17-s − 1/5·25-s + 0.742·29-s − 0.328·37-s − 1.56·41-s − 0.549·53-s + 1.53·61-s − 0.992·65-s + 1.87·73-s + 0.433·85-s − 1.05·89-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(225.369\)
Root analytic conductor: \(15.0123\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.57268345741849, −14.74602721545381, −14.47284946833997, −13.86401651118500, −13.43905118033363, −12.85554059207646, −12.20044949731823, −11.91928474426002, −11.15743270368973, −10.49767334691755, −9.921348113057571, −9.720781151699097, −9.044430346825993, −8.311696759901305, −7.863457104176213, −7.037184155741507, −6.650418683188425, −5.945380017267005, −5.234092500338557, −4.968573003468174, −4.043981168507767, −3.299908152424339, −2.529350584953931, −1.973866429593405, −1.131824779317929, 0, 1.131824779317929, 1.973866429593405, 2.529350584953931, 3.299908152424339, 4.043981168507767, 4.968573003468174, 5.234092500338557, 5.945380017267005, 6.650418683188425, 7.037184155741507, 7.863457104176213, 8.311696759901305, 9.044430346825993, 9.720781151699097, 9.921348113057571, 10.49767334691755, 11.15743270368973, 11.91928474426002, 12.20044949731823, 12.85554059207646, 13.43905118033363, 13.86401651118500, 14.47284946833997, 14.74602721545381, 15.57268345741849

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