Properties

Label 2-154560-1.1-c1-0-147
Degree $2$
Conductor $154560$
Sign $-1$
Analytic cond. $1234.16$
Root an. cond. $35.1307$
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 + 9-s − 2·11-s + 4·13-s + 15-s − 6·17-s + 2·19-s − 21-s + 23-s + 25-s + 27-s − 6·29-s + 2·31-s − 2·33-s − 35-s − 10·37-s + 4·39-s − 6·41-s + 4·43-s + 45-s + 8·47-s + 49-s − 6·51-s − 2·55-s + 2·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s + 0.258·15-s − 1.45·17-s + 0.458·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.359·31-s − 0.348·33-s − 0.169·35-s − 1.64·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.840·51-s − 0.269·55-s + 0.264·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154560\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(1234.16\)
Root analytic conductor: \(35.1307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 154560,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.46977998429466, −13.23222605316512, −12.79033312660222, −12.21835887673729, −11.61508760508492, −10.98394343070590, −10.79098479775555, −10.07842954113379, −9.815428623512668, −8.988834166268986, −8.797312529625848, −8.495369302104197, −7.624982701426969, −7.267138329308590, −6.725993847068032, −6.173496638425949, −5.715788307515787, −5.083268933413171, −4.573000557281198, −3.792864833451213, −3.488744938142402, −2.792479604983207, −2.181423177694044, −1.724260240299504, −0.8883751522166843, 0, 0.8883751522166843, 1.724260240299504, 2.181423177694044, 2.792479604983207, 3.488744938142402, 3.792864833451213, 4.573000557281198, 5.083268933413171, 5.715788307515787, 6.173496638425949, 6.725993847068032, 7.267138329308590, 7.624982701426969, 8.495369302104197, 8.797312529625848, 8.988834166268986, 9.815428623512668, 10.07842954113379, 10.79098479775555, 10.98394343070590, 11.61508760508492, 12.21835887673729, 12.79033312660222, 13.23222605316512, 13.46977998429466

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