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Properties

Label 2-185e2-1.1-c1-0-8
Degree $2$
Conductor $34225$
Sign $-1$
Analytic cond. $273.288$
Root an. cond. $16.5314$
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 + 3·3-s + 2·4-s − 6·6-s + 7-s + 6·9-s − 5·11-s + 6·12-s − 2·13-s − 2·14-s − 4·16-s − 12·18-s + 3·21-s + 10·22-s + 2·23-s + 4·26-s + 9·27-s + 2·28-s − 6·29-s + 4·31-s + 8·32-s − 15·33-s + 12·36-s − 6·39-s − 9·41-s − 6·42-s + 2·43-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s + 4-s − 2.44·6-s + 0.377·7-s + 2·9-s − 1.50·11-s + 1.73·12-s − 0.554·13-s − 0.534·14-s − 16-s − 2.82·18-s + 0.654·21-s + 2.13·22-s + 0.417·23-s + 0.784·26-s + 1.73·27-s + 0.377·28-s − 1.11·29-s + 0.718·31-s + 1.41·32-s − 2.61·33-s + 2·36-s − 0.960·39-s − 1.40·41-s − 0.925·42-s + 0.304·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(34225\)    =    \(5^{2} \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(273.288\)
Root analytic conductor: \(16.5314\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 34225,\ (\ :1/2),\ -1)\)

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 \( 1 \)
37 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 - p T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−15.38332815413334, −14.79876956505917, −14.26753474674523, −13.59041395500666, −13.35149762478085, −12.75813308784845, −12.08568022574822, −11.22702567378086, −10.74404485652910, −10.11015786669984, −9.863050165904891, −9.204167625888170, −8.772973053818475, −8.238111601065694, −7.878134393030590, −7.375651259153501, −7.073845592528331, −6.051168246591496, −4.996908218077302, −4.672645125050427, −3.686354422774425, −3.031191200693463, −2.287126256220979, −2.016121581401957, −1.071012726527684, 0, 1.071012726527684, 2.016121581401957, 2.287126256220979, 3.031191200693463, 3.686354422774425, 4.672645125050427, 4.996908218077302, 6.051168246591496, 7.073845592528331, 7.375651259153501, 7.878134393030590, 8.238111601065694, 8.772973053818475, 9.204167625888170, 9.863050165904891, 10.11015786669984, 10.74404485652910, 11.22702567378086, 12.08568022574822, 12.75813308784845, 13.35149762478085, 13.59041395500666, 14.26753474674523, 14.79876956505917, 15.38332815413334

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