Properties

Label 2-168175-1.1-c1-0-16
Degree $2$
Conductor $168175$
Sign $-1$
Analytic cond. $1342.88$
Root an. cond. $36.6453$
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 − 2·9-s + 3·11-s − 2·12-s − 13-s − 2·14-s − 4·16-s − 7·17-s − 4·18-s + 21-s + 6·22-s − 6·23-s − 2·26-s + 5·27-s − 2·28-s + 5·29-s − 8·32-s − 3·33-s − 14·34-s − 4·36-s − 2·37-s + 39-s + 2·41-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s − 0.277·13-s − 0.534·14-s − 16-s − 1.69·17-s − 0.942·18-s + 0.218·21-s + 1.27·22-s − 1.25·23-s − 0.392·26-s + 0.962·27-s − 0.377·28-s + 0.928·29-s − 1.41·32-s − 0.522·33-s − 2.40·34-s − 2/3·36-s − 0.328·37-s + 0.160·39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168175\)    =    \(5^{2} \cdot 7 \cdot 31^{2}\)
Sign: $-1$
Analytic conductor: \(1342.88\)
Root analytic conductor: \(36.6453\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 168175,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 \)
7 \( 1 + T \)
31 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.46116018916633, −13.03175738654772, −12.48520446588452, −12.11609264427755, −11.66462603295253, −11.39190961340680, −10.85154207316904, −10.32657497479134, −9.655653298510648, −9.185681394250124, −8.633897511573177, −8.298388837164532, −7.383351271980296, −6.787701770092888, −6.432170695686167, −6.135686450787227, −5.577152203402406, −5.037668883453409, −4.475775364333045, −4.125946217946944, −3.566868563902065, −2.900190651857540, −2.384022188004677, −1.821773259014984, −0.6986669036867241, 0, 0.6986669036867241, 1.821773259014984, 2.384022188004677, 2.900190651857540, 3.566868563902065, 4.125946217946944, 4.475775364333045, 5.037668883453409, 5.577152203402406, 6.135686450787227, 6.432170695686167, 6.787701770092888, 7.383351271980296, 8.298388837164532, 8.633897511573177, 9.185681394250124, 9.655653298510648, 10.32657497479134, 10.85154207316904, 11.39190961340680, 11.66462603295253, 12.11609264427755, 12.48520446588452, 13.03175738654772, 13.46116018916633

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