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Properties

Label 2-169050-1.1-c1-0-165
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 $1$

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 − 3·13-s + 16-s − 4·17-s − 18-s − 4·22-s − 23-s − 24-s + 3·26-s + 27-s + 3·29-s + 6·31-s − 32-s + 4·33-s + 4·34-s + 36-s + 9·37-s − 3·39-s − 9·41-s + 3·43-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 − 0.832·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.852·22-s − 0.208·23-s − 0.204·24-s + 0.588·26-s + 0.192·27-s + 0.557·29-s + 1.07·31-s − 0.176·32-s + 0.696·33-s + 0.685·34-s + 1/6·36-s + 1.47·37-s − 0.480·39-s − 1.40·41-s + 0.457·43-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: \(1\)
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 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.54552981609806, −13.05936448016574, −12.32097391353146, −12.00503448443249, −11.60493670408693, −11.09250963820530, −10.41219477000119, −10.10217651507123, −9.530851104175839, −9.121931128652432, −8.820203516872143, −8.160668850718263, −7.810523499830217, −7.228717679102179, −6.671434899233543, −6.363688094293234, −5.832849413679000, −4.775959866265424, −4.608089072901899, −3.929526733295887, −3.223198999579038, −2.726147975835730, −2.115127606268087, −1.539892193087567, −0.8709072711053856, 0, 0.8709072711053856, 1.539892193087567, 2.115127606268087, 2.726147975835730, 3.223198999579038, 3.929526733295887, 4.608089072901899, 4.775959866265424, 5.832849413679000, 6.363688094293234, 6.671434899233543, 7.228717679102179, 7.810523499830217, 8.160668850718263, 8.820203516872143, 9.121931128652432, 9.530851104175839, 10.10217651507123, 10.41219477000119, 11.09250963820530, 11.60493670408693, 12.00503448443249, 12.32097391353146, 13.05936448016574, 13.54552981609806

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