Properties

Label 2-169050-1.1-c1-0-188
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 + 3·11-s − 12-s + 4·13-s + 16-s − 6·17-s + 18-s + 2·19-s + 3·22-s + 23-s − 24-s + 4·26-s − 27-s + 6·29-s − 4·31-s + 32-s − 3·33-s − 6·34-s + 36-s − 11·37-s + 2·38-s − 4·39-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 + 0.904·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.458·19-s + 0.639·22-s + 0.208·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.522·33-s − 1.02·34-s + 1/6·36-s − 1.80·37-s + 0.324·38-s − 0.640·39-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 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
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

−13.53691548842918, −12.91037135409653, −12.54600114173356, −12.06876451007066, −11.55471202685113, −11.18471514862169, −10.77555049548827, −10.39965107711061, −9.647000184863396, −9.130582896965644, −8.718372312231710, −8.195529208122386, −7.489678836195200, −6.867731381026894, −6.579687272646522, −6.192949829521586, −5.596756877408976, −5.005601214854638, −4.590396951121301, −3.942811514532196, −3.591481655351534, −2.940317698083001, −2.128408642480966, −1.543080913767647, −0.9709365577894886, 0, 0.9709365577894886, 1.543080913767647, 2.128408642480966, 2.940317698083001, 3.591481655351534, 3.942811514532196, 4.590396951121301, 5.005601214854638, 5.596756877408976, 6.192949829521586, 6.579687272646522, 6.867731381026894, 7.489678836195200, 8.195529208122386, 8.718372312231710, 9.130582896965644, 9.647000184863396, 10.39965107711061, 10.77555049548827, 11.18471514862169, 11.55471202685113, 12.06876451007066, 12.54600114173356, 12.91037135409653, 13.53691548842918

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