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Properties

Label 2-63525-1.1-c1-0-22
Degree $2$
Conductor $63525$
Sign $-1$
Analytic cond. $507.249$
Root an. cond. $22.5222$
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  − 2·2-s − 3-s + 2·4-s + 2·6-s + 7-s + 9-s − 2·12-s − 6·13-s − 2·14-s − 4·16-s − 7·17-s − 2·18-s + 5·19-s − 21-s + 23-s + 12·26-s − 27-s + 2·28-s + 5·29-s − 8·31-s + 8·32-s + 14·34-s + 2·36-s + 2·37-s − 10·38-s + 6·39-s − 12·41-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.577·12-s − 1.66·13-s − 0.534·14-s − 16-s − 1.69·17-s − 0.471·18-s + 1.14·19-s − 0.218·21-s + 0.208·23-s + 2.35·26-s − 0.192·27-s + 0.377·28-s + 0.928·29-s − 1.43·31-s + 1.41·32-s + 2.40·34-s + 1/3·36-s + 0.328·37-s − 1.62·38-s + 0.960·39-s − 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63525\)    =    \(3 \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(507.249\)
Root analytic conductor: \(22.5222\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 63525,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−14.74539208089668, −13.89611580540989, −13.48500722616599, −12.99566596102188, −12.18485697729022, −11.79627770262787, −11.42567926810923, −10.76767687763461, −10.42548776701373, −9.852012562626177, −9.368995400471281, −9.049096599486922, −8.203459569596387, −7.988618153292165, −7.157254473274779, −6.901331887298360, −6.486678422809499, −5.397268786552735, −4.925008320398704, −4.652475674353361, −3.695124056798726, −2.814782010891988, −2.036290679051076, −1.651174457442897, −0.6279318758735553, 0, 0.6279318758735553, 1.651174457442897, 2.036290679051076, 2.814782010891988, 3.695124056798726, 4.652475674353361, 4.925008320398704, 5.397268786552735, 6.486678422809499, 6.901331887298360, 7.157254473274779, 7.988618153292165, 8.203459569596387, 9.049096599486922, 9.368995400471281, 9.852012562626177, 10.42548776701373, 10.76767687763461, 11.42567926810923, 11.79627770262787, 12.18485697729022, 12.99566596102188, 13.48500722616599, 13.89611580540989, 14.74539208089668

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