Properties

Label 2-75712-1.1-c1-0-59
Degree $2$
Conductor $75712$
Sign $-1$
Analytic cond. $604.563$
Root an. cond. $24.5878$
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  + 5-s − 7-s − 3·9-s + 2·11-s − 3·17-s + 6·19-s − 4·23-s − 4·25-s + 7·29-s + 4·31-s − 35-s − 9·37-s + 9·41-s − 10·43-s − 3·45-s − 2·47-s + 49-s − 9·53-s + 2·55-s − 14·59-s + 5·61-s + 3·63-s + 8·67-s + 10·71-s − 7·73-s − 2·77-s + 2·79-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 9-s + 0.603·11-s − 0.727·17-s + 1.37·19-s − 0.834·23-s − 4/5·25-s + 1.29·29-s + 0.718·31-s − 0.169·35-s − 1.47·37-s + 1.40·41-s − 1.52·43-s − 0.447·45-s − 0.291·47-s + 1/7·49-s − 1.23·53-s + 0.269·55-s − 1.82·59-s + 0.640·61-s + 0.377·63-s + 0.977·67-s + 1.18·71-s − 0.819·73-s − 0.227·77-s + 0.225·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75712\)    =    \(2^{6} \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(604.563\)
Root analytic conductor: \(24.5878\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 75712,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.22909648685045, −13.75116496102445, −13.60624749977999, −12.78845919640548, −12.19296345831926, −11.82181020552207, −11.43318138141217, −10.81599220686073, −10.18661417312130, −9.764884914192902, −9.253099394156371, −8.842613949819286, −8.131349975796546, −7.835894770230455, −6.975168111339528, −6.404001406521185, −6.165055548989551, −5.457261783224381, −4.939039738226868, −4.309488206199336, −3.445518696775843, −3.156342821290636, −2.346868827757274, −1.753794686686920, −0.8739813041508053, 0, 0.8739813041508053, 1.753794686686920, 2.346868827757274, 3.156342821290636, 3.445518696775843, 4.309488206199336, 4.939039738226868, 5.457261783224381, 6.165055548989551, 6.404001406521185, 6.975168111339528, 7.835894770230455, 8.131349975796546, 8.842613949819286, 9.253099394156371, 9.764884914192902, 10.18661417312130, 10.81599220686073, 11.43318138141217, 11.82181020552207, 12.19296345831926, 12.78845919640548, 13.60624749977999, 13.75116496102445, 14.22909648685045

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