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Properties

Label 2-169050-1.1-c1-0-226
Degree $2$
Conductor $169050$
Sign $1$
Analytic cond. $1349.87$
Root an. cond. $36.7405$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 4·11-s − 12-s + 13-s + 16-s − 18-s + 19-s + 4·22-s − 23-s + 24-s − 26-s − 27-s − 6·29-s + 4·31-s − 32-s + 4·33-s + 36-s − 11·37-s − 38-s − 39-s − 6·41-s − 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s − 0.235·18-s + 0.229·19-s + 0.852·22-s − 0.208·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.696·33-s + 1/6·36-s − 1.80·37-s − 0.162·38-s − 0.160·39-s − 0.937·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1349.87\)
Root analytic conductor: \(36.7405\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 169050,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
23 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.61607495256776, −13.13883429119177, −12.74342976192967, −12.12383503704302, −11.71027840723995, −11.36498986901098, −10.64537713333531, −10.38414904413492, −10.11542212887037, −9.390817230925440, −8.954303825786094, −8.373360355757011, −7.953862525674415, −7.442685881597119, −6.967599915025088, −6.478343529745732, −5.868411251523663, −5.310648420220627, −5.077882736607266, −4.242152439085104, −3.616865275897245, −3.009665841529489, −2.395517879050228, −1.687685884595334, −1.174310149100215, 0, 0, 1.174310149100215, 1.687685884595334, 2.395517879050228, 3.009665841529489, 3.616865275897245, 4.242152439085104, 5.077882736607266, 5.310648420220627, 5.868411251523663, 6.478343529745732, 6.967599915025088, 7.442685881597119, 7.953862525674415, 8.373360355757011, 8.954303825786094, 9.390817230925440, 10.11542212887037, 10.38414904413492, 10.64537713333531, 11.36498986901098, 11.71027840723995, 12.12383503704302, 12.74342976192967, 13.13883429119177, 13.61607495256776

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