Properties

Label 2-115920-1.1-c1-0-8
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 − 6·13-s + 6·17-s − 4·19-s + 23-s + 25-s + 6·29-s + 35-s − 10·37-s − 2·41-s + 8·43-s + 8·47-s + 49-s + 10·53-s − 4·59-s + 14·61-s + 6·65-s + 16·67-s − 12·71-s − 14·73-s + 8·79-s + 4·83-s − 6·85-s − 14·89-s + 6·91-s + 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.66·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.11·29-s + 0.169·35-s − 1.64·37-s − 0.312·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s + 1.37·53-s − 0.520·59-s + 1.79·61-s + 0.744·65-s + 1.95·67-s − 1.42·71-s − 1.63·73-s + 0.900·79-s + 0.439·83-s − 0.650·85-s − 1.48·89-s + 0.628·91-s + 0.410·95-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

\(L(1)\) \(\approx\) \(1.531212147\)
\(L(\frac12)\) \(\approx\) \(1.531212147\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.74552154928391, −12.88571301303916, −12.57437695324418, −12.18098096053473, −11.84717706173591, −11.25424795544273, −10.45028402837857, −10.20295934025527, −9.902900429665964, −9.099044618416651, −8.715134701259381, −8.171478193179430, −7.497947728344323, −7.227239999076281, −6.730223312327865, −6.044003342797308, −5.385464968170115, −5.057232998962117, −4.331285540395110, −3.846768715076992, −3.184156839191224, −2.593074559092019, −2.103034713717097, −1.084202776230313, −0.4229984540891425, 0.4229984540891425, 1.084202776230313, 2.103034713717097, 2.593074559092019, 3.184156839191224, 3.846768715076992, 4.331285540395110, 5.057232998962117, 5.385464968170115, 6.044003342797308, 6.730223312327865, 7.227239999076281, 7.497947728344323, 8.171478193179430, 8.715134701259381, 9.099044618416651, 9.902900429665964, 10.20295934025527, 10.45028402837857, 11.25424795544273, 11.84717706173591, 12.18098096053473, 12.57437695324418, 12.88571301303916, 13.74552154928391

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