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Properties

Label 2-115920-1.1-c1-0-96
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 + 2·13-s + 4·19-s − 23-s + 25-s + 4·31-s + 35-s + 2·37-s − 12·41-s + 4·43-s + 12·47-s + 49-s + 12·59-s + 14·61-s − 2·65-s + 4·67-s − 12·71-s − 4·73-s − 8·79-s − 6·89-s − 2·91-s − 4·95-s − 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 0.554·13-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.718·31-s + 0.169·35-s + 0.328·37-s − 1.87·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 1.56·59-s + 1.79·61-s − 0.248·65-s + 0.488·67-s − 1.42·71-s − 0.468·73-s − 0.900·79-s − 0.635·89-s − 0.209·91-s − 0.410·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: Trivial
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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.81733031955514, −13.23830508889071, −13.06520293504269, −12.24123443368461, −11.93281608207041, −11.48453860530154, −11.02696348509936, −10.28283031976977, −10.06665620761272, −9.486856403185328, −8.747278303522331, −8.574480842461458, −7.893105137386480, −7.365416201408877, −6.885775996935125, −6.410080463295770, −5.688431127966893, −5.346776658073557, −4.648773469570866, −3.854888409404952, −3.748633798841891, −2.830619233722500, −2.465232913664965, −1.428827988772603, −0.8923026544962361, 0, 0.8923026544962361, 1.428827988772603, 2.465232913664965, 2.830619233722500, 3.748633798841891, 3.854888409404952, 4.648773469570866, 5.346776658073557, 5.688431127966893, 6.410080463295770, 6.885775996935125, 7.365416201408877, 7.893105137386480, 8.574480842461458, 8.747278303522331, 9.486856403185328, 10.06665620761272, 10.28283031976977, 11.02696348509936, 11.48453860530154, 11.93281608207041, 12.24123443368461, 13.06520293504269, 13.23830508889071, 13.81733031955514

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