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Properties

Label 2-121968-1.1-c1-0-136
Degree $2$
Conductor $121968$
Sign $-1$
Analytic cond. $973.919$
Root an. cond. $31.2076$
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  + 3·5-s + 7-s − 7·13-s + 3·17-s − 2·19-s − 4·23-s + 4·25-s + 7·29-s + 10·31-s + 3·35-s + 37-s − 5·41-s + 6·43-s − 6·47-s + 49-s + 5·53-s − 6·59-s − 10·61-s − 21·65-s + 8·67-s − 10·71-s − 10·73-s − 2·79-s + 16·83-s + 9·85-s − 3·89-s − 7·91-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s − 1.94·13-s + 0.727·17-s − 0.458·19-s − 0.834·23-s + 4/5·25-s + 1.29·29-s + 1.79·31-s + 0.507·35-s + 0.164·37-s − 0.780·41-s + 0.914·43-s − 0.875·47-s + 1/7·49-s + 0.686·53-s − 0.781·59-s − 1.28·61-s − 2.60·65-s + 0.977·67-s − 1.18·71-s − 1.17·73-s − 0.225·79-s + 1.75·83-s + 0.976·85-s − 0.317·89-s − 0.733·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121968\)    =    \(2^{4} \cdot 3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(973.919\)
Root analytic conductor: \(31.2076\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 121968,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−13.87185329230204, −13.50305001107202, −12.72357173018484, −12.33510626746431, −11.97812215477001, −11.52038284376456, −10.66229272728772, −10.20030717619936, −9.940035714381190, −9.654224322233990, −8.909157077976425, −8.452861930261018, −7.770212157759800, −7.481824363185867, −6.667485243892212, −6.308588831463295, −5.771352513169232, −5.203297250209878, −4.649329830420939, −4.403881282841402, −3.318024858664244, −2.650333766801629, −2.346792507488647, −1.663028371413091, −0.9866990523760759, 0, 0.9866990523760759, 1.663028371413091, 2.346792507488647, 2.650333766801629, 3.318024858664244, 4.403881282841402, 4.649329830420939, 5.203297250209878, 5.771352513169232, 6.308588831463295, 6.667485243892212, 7.481824363185867, 7.770212157759800, 8.452861930261018, 8.909157077976425, 9.654224322233990, 9.940035714381190, 10.20030717619936, 10.66229272728772, 11.52038284376456, 11.97812215477001, 12.33510626746431, 12.72357173018484, 13.50305001107202, 13.87185329230204

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