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Properties

Label 2-76230-1.1-c1-0-100
Degree $2$
Conductor $76230$
Sign $-1$
Analytic cond. $608.699$
Root an. cond. $24.6718$
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 + 5-s + 7-s − 8-s − 10-s + 2·13-s − 14-s + 16-s − 6·17-s + 20-s + 8·23-s + 25-s − 2·26-s + 28-s + 10·29-s − 8·31-s − 32-s + 6·34-s + 35-s + 2·37-s − 40-s − 2·41-s − 8·43-s − 8·46-s − 4·47-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.223·20-s + 1.66·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 1.85·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.328·37-s − 0.158·40-s − 0.312·41-s − 1.21·43-s − 1.17·46-s − 0.583·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76230\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(608.699\)
Root analytic conductor: \(24.6718\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 76230,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.32596367103330, −13.76035939885268, −13.31681457288451, −12.76638902974875, −12.38877506257450, −11.51539579693375, −11.20606364658440, −10.81750801107854, −10.36371267268836, −9.583870617797076, −9.313484722276844, −8.640769642895045, −8.388686438312088, −7.795485378818042, −6.904242610153530, −6.739582134807513, −6.244357796499513, −5.366889365308611, −4.959761955008611, −4.357654355351589, −3.499618423307866, −2.902544578845552, −2.241639205990518, −1.578906182817533, −0.9784127581726939, 0, 0.9784127581726939, 1.578906182817533, 2.241639205990518, 2.902544578845552, 3.499618423307866, 4.357654355351589, 4.959761955008611, 5.366889365308611, 6.244357796499513, 6.739582134807513, 6.904242610153530, 7.795485378818042, 8.388686438312088, 8.640769642895045, 9.313484722276844, 9.583870617797076, 10.36371267268836, 10.81750801107854, 11.20606364658440, 11.51539579693375, 12.38877506257450, 12.76638902974875, 13.31681457288451, 13.76035939885268, 14.32596367103330

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