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Properties

Label 2-616e2-1.1-c1-0-234
Degree $2$
Conductor $379456$
Sign $-1$
Analytic cond. $3029.97$
Root an. cond. $55.0451$
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  + 2·3-s + 4·5-s + 9-s − 2·13-s + 8·15-s + 2·17-s − 8·19-s + 11·25-s − 4·27-s − 6·29-s − 2·31-s + 10·37-s − 4·39-s + 10·41-s − 8·43-s + 4·45-s − 6·47-s + 4·51-s + 6·53-s − 16·57-s − 6·59-s + 2·61-s − 8·65-s + 12·67-s − 8·71-s − 2·73-s + 22·75-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 1/3·9-s − 0.554·13-s + 2.06·15-s + 0.485·17-s − 1.83·19-s + 11/5·25-s − 0.769·27-s − 1.11·29-s − 0.359·31-s + 1.64·37-s − 0.640·39-s + 1.56·41-s − 1.21·43-s + 0.596·45-s − 0.875·47-s + 0.560·51-s + 0.824·53-s − 2.11·57-s − 0.781·59-s + 0.256·61-s − 0.992·65-s + 1.46·67-s − 0.949·71-s − 0.234·73-s + 2.54·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(379456\)    =    \(2^{6} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(3029.97\)
Root analytic conductor: \(55.0451\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 379456,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−12.80837640307117, −12.61855658511363, −11.84049674206269, −11.16706481114421, −10.84916759464220, −10.28849234416822, −9.819326529386208, −9.529962714060682, −9.104125297587246, −8.814155995128895, −8.049283018825183, −7.951086954925903, −7.187533058550086, −6.677172951828200, −6.184721600881425, −5.795706482210066, −5.341472926055522, −4.717582218639214, −4.190783891365980, −3.575467269612320, −2.991013529996282, −2.358963516285134, −2.218489219190576, −1.712422777409354, −0.9762588060345908, 0, 0.9762588060345908, 1.712422777409354, 2.218489219190576, 2.358963516285134, 2.991013529996282, 3.575467269612320, 4.190783891365980, 4.717582218639214, 5.341472926055522, 5.795706482210066, 6.184721600881425, 6.677172951828200, 7.187533058550086, 7.951086954925903, 8.049283018825183, 8.814155995128895, 9.104125297587246, 9.529962714060682, 9.819326529386208, 10.28849234416822, 10.84916759464220, 11.16706481114421, 11.84049674206269, 12.61855658511363, 12.80837640307117

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