Properties

Label 2-115920-1.1-c1-0-37
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 6·11-s − 6·17-s + 8·19-s + 23-s + 25-s + 6·29-s + 6·31-s + 35-s + 8·37-s + 10·41-s + 8·43-s + 8·47-s + 49-s + 4·53-s + 6·55-s − 10·59-s + 2·61-s + 4·67-s + 4·73-s + 6·77-s + 2·79-s + 4·83-s + 6·85-s + 10·89-s − 8·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.80·11-s − 1.45·17-s + 1.83·19-s + 0.208·23-s + 1/5·25-s + 1.11·29-s + 1.07·31-s + 0.169·35-s + 1.31·37-s + 1.56·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.549·53-s + 0.809·55-s − 1.30·59-s + 0.256·61-s + 0.488·67-s + 0.468·73-s + 0.683·77-s + 0.225·79-s + 0.439·83-s + 0.650·85-s + 1.05·89-s − 0.820·95-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: \(0\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.188711032\)
\(L(\frac12)\) \(\approx\) \(2.188711032\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.59709978338266, −13.12475878051894, −12.69947853628569, −12.14975979734969, −11.70804808634519, −11.08618600682667, −10.67986492013961, −10.36205085037263, −9.489118423115922, −9.397651853113957, −8.628981637722258, −8.045007228280076, −7.669609992910002, −7.266411094264758, −6.633181567959952, −5.975817877260350, −5.570525354677049, −4.800136553997364, −4.547289454493772, −3.846085620337352, −2.958894566583857, −2.719609895677669, −2.199735441036006, −0.8986919436092846, −0.5885971726833580, 0.5885971726833580, 0.8986919436092846, 2.199735441036006, 2.719609895677669, 2.958894566583857, 3.846085620337352, 4.547289454493772, 4.800136553997364, 5.570525354677049, 5.975817877260350, 6.633181567959952, 7.266411094264758, 7.669609992910002, 8.045007228280076, 8.628981637722258, 9.397651853113957, 9.489118423115922, 10.36205085037263, 10.67986492013961, 11.08618600682667, 11.70804808634519, 12.14975979734969, 12.69947853628569, 13.12475878051894, 13.59709978338266

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