Properties

Label 2-75712-1.1-c1-0-42
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  + 3-s − 5-s + 7-s − 2·9-s − 4·11-s − 15-s − 6·17-s − 4·19-s + 21-s + 23-s − 4·25-s − 5·27-s + 6·29-s + 6·31-s − 4·33-s − 35-s + 4·37-s − 8·41-s + 4·43-s + 2·45-s + 6·47-s + 49-s − 6·51-s − 4·53-s + 4·55-s − 4·57-s − 3·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.20·11-s − 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.218·21-s + 0.208·23-s − 4/5·25-s − 0.962·27-s + 1.11·29-s + 1.07·31-s − 0.696·33-s − 0.169·35-s + 0.657·37-s − 1.24·41-s + 0.609·43-s + 0.298·45-s + 0.875·47-s + 1/7·49-s − 0.840·51-s − 0.549·53-s + 0.539·55-s − 0.529·57-s − 0.390·59-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: $\chi_{75712} (1, \cdot )$
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 \( 1 - T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−14.20291933414404, −13.70607195954211, −13.53124790747662, −12.85014243673903, −12.34609659222033, −11.77481960985165, −11.26872884390982, −10.78425933388771, −10.45386351852090, −9.701839883895229, −9.148326358186684, −8.580368082596321, −8.150724370804503, −7.934197569853234, −7.244623762938414, −6.456304236715471, −6.183960278223667, −5.292879600210023, −4.810201707678349, −4.297697394306653, −3.639163611586383, −2.914050119184296, −2.352564748228897, −2.021103546206564, −0.7517080898745030, 0, 0.7517080898745030, 2.021103546206564, 2.352564748228897, 2.914050119184296, 3.639163611586383, 4.297697394306653, 4.810201707678349, 5.292879600210023, 6.183960278223667, 6.456304236715471, 7.244623762938414, 7.934197569853234, 8.150724370804503, 8.580368082596321, 9.148326358186684, 9.701839883895229, 10.45386351852090, 10.78425933388771, 11.26872884390982, 11.77481960985165, 12.34609659222033, 12.85014243673903, 13.53124790747662, 13.70607195954211, 14.20291933414404

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