Properties

Label 2-115920-1.1-c1-0-126
Degree $2$
Conductor $115920$
Sign $-1$
Analytic cond. $925.625$
Root an. cond. $30.4240$
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 − 2·11-s + 2·17-s + 2·19-s − 23-s + 25-s + 10·29-s + 2·31-s + 35-s − 2·37-s − 2·41-s + 4·43-s + 49-s − 8·53-s − 2·55-s + 12·59-s − 2·61-s + 4·67-s − 6·71-s − 16·73-s − 2·77-s − 4·79-s + 4·83-s + 2·85-s + 10·89-s + 2·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.603·11-s + 0.485·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s + 1.85·29-s + 0.359·31-s + 0.169·35-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 1/7·49-s − 1.09·53-s − 0.269·55-s + 1.56·59-s − 0.256·61-s + 0.488·67-s − 0.712·71-s − 1.87·73-s − 0.227·77-s − 0.450·79-s + 0.439·83-s + 0.216·85-s + 1.05·89-s + 0.205·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(925.625\)
Root analytic conductor: \(30.4240\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{115920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.88301803556912, −13.35785082703416, −12.97624030479723, −12.29686802019155, −11.97390027652550, −11.48736582820294, −10.84260155728579, −10.34282736140023, −10.06501049954160, −9.519839850785635, −8.864507138658274, −8.435453910334913, −7.906785371784337, −7.481816418427529, −6.796916310824036, −6.350369950955680, −5.715691266689715, −5.235713936698639, −4.769486137710166, −4.182833609324866, −3.444833903935260, −2.767339055210451, −2.416622823077538, −1.472577532038171, −1.022384935679606, 0, 1.022384935679606, 1.472577532038171, 2.416622823077538, 2.767339055210451, 3.444833903935260, 4.182833609324866, 4.769486137710166, 5.235713936698639, 5.715691266689715, 6.350369950955680, 6.796916310824036, 7.481816418427529, 7.906785371784337, 8.435453910334913, 8.864507138658274, 9.519839850785635, 10.06501049954160, 10.34282736140023, 10.84260155728579, 11.48736582820294, 11.97390027652550, 12.29686802019155, 12.97624030479723, 13.35785082703416, 13.88301803556912

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