Properties

Label 2-101150-1.1-c1-0-33
Degree $2$
Conductor $101150$
Sign $-1$
Analytic cond. $807.686$
Root an. cond. $28.4198$
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-s + 4-s − 7-s − 8-s − 3·9-s − 4·11-s + 6·13-s + 14-s + 16-s + 3·18-s + 4·22-s − 6·26-s − 28-s − 6·29-s − 8·31-s − 32-s − 3·36-s − 10·37-s − 2·41-s − 4·43-s − 4·44-s − 8·47-s + 49-s + 6·52-s + 2·53-s + 56-s + 6·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 9-s − 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.707·18-s + 0.852·22-s − 1.17·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1/2·36-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.603·44-s − 1.16·47-s + 1/7·49-s + 0.832·52-s + 0.274·53-s + 0.133·56-s + 0.787·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(101150\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(807.686\)
Root analytic conductor: \(28.4198\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 101150,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
17 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.73861738998774, −13.66527486572513, −12.87768478162605, −12.67225987203376, −11.90846904150114, −11.28207429611528, −11.00940894107766, −10.68527481251396, −10.02967133409269, −9.491596104472005, −8.992908627962992, −8.396169656233916, −8.245424059381933, −7.614379083133301, −6.903174094456593, −6.518312879597412, −5.841817830324479, −5.370916474048445, −5.061080838504499, −3.700239814210280, −3.625709909790842, −2.948156658945282, −2.135816737401968, −1.693926953094215, −0.6537340265133293, 0, 0.6537340265133293, 1.693926953094215, 2.135816737401968, 2.948156658945282, 3.625709909790842, 3.700239814210280, 5.061080838504499, 5.370916474048445, 5.841817830324479, 6.518312879597412, 6.903174094456593, 7.614379083133301, 8.245424059381933, 8.396169656233916, 8.992908627962992, 9.491596104472005, 10.02967133409269, 10.68527481251396, 11.00940894107766, 11.28207429611528, 11.90846904150114, 12.67225987203376, 12.87768478162605, 13.66527486572513, 13.73861738998774

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