Properties

Label 2-115920-1.1-c1-0-121
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 + 6·17-s + 4·19-s + 23-s + 25-s − 2·29-s − 10·31-s + 35-s − 8·37-s − 6·41-s + 12·43-s + 4·47-s + 49-s + 8·53-s − 2·55-s − 10·59-s + 6·61-s + 8·67-s + 4·71-s − 16·73-s − 2·77-s − 10·79-s − 12·83-s + 6·85-s + 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.603·11-s + 1.45·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s − 1.79·31-s + 0.169·35-s − 1.31·37-s − 0.937·41-s + 1.82·43-s + 0.583·47-s + 1/7·49-s + 1.09·53-s − 0.269·55-s − 1.30·59-s + 0.768·61-s + 0.977·67-s + 0.474·71-s − 1.87·73-s − 0.227·77-s − 1.12·79-s − 1.31·83-s + 0.650·85-s + 0.635·89-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 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.94756485885724, −13.32150716995425, −12.92589481099509, −12.31045714073213, −12.07058546645728, −11.33533647378241, −10.94529971564378, −10.38948989728338, −9.961302319611004, −9.544384484493789, −8.774605153854453, −8.647975993184194, −7.717655481464691, −7.357127195732895, −7.158600375453582, −6.152661810648403, −5.613234370466562, −5.372427529732366, −4.876141506285254, −3.990502464375432, −3.503559494232171, −2.918177474012922, −2.242637491879188, −1.552281449621212, −0.9967402506326371, 0, 0.9967402506326371, 1.552281449621212, 2.242637491879188, 2.918177474012922, 3.503559494232171, 3.990502464375432, 4.876141506285254, 5.372427529732366, 5.613234370466562, 6.152661810648403, 7.158600375453582, 7.357127195732895, 7.717655481464691, 8.647975993184194, 8.774605153854453, 9.544384484493789, 9.961302319611004, 10.38948989728338, 10.94529971564378, 11.33533647378241, 12.07058546645728, 12.31045714073213, 12.92589481099509, 13.32150716995425, 13.94756485885724

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