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Properties

Label 2-115920-1.1-c1-0-104
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 − 2·17-s + 4·19-s + 23-s + 25-s + 6·29-s − 8·31-s − 35-s + 6·37-s − 2·41-s − 8·43-s + 49-s + 10·53-s + 12·59-s − 2·61-s − 2·65-s + 4·71-s − 14·73-s − 8·79-s + 12·83-s + 2·85-s − 6·89-s + 2·91-s − 4·95-s − 10·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.169·35-s + 0.986·37-s − 0.312·41-s − 1.21·43-s + 1/7·49-s + 1.37·53-s + 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.474·71-s − 1.63·73-s − 0.900·79-s + 1.31·83-s + 0.216·85-s − 0.635·89-s + 0.209·91-s − 0.410·95-s − 1.01·97-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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−13.76441712219253, −13.33231488925419, −13.01546655799335, −12.26703252273047, −11.87005641639079, −11.44323775900183, −10.99754575696697, −10.53302218808692, −9.924145655805277, −9.466782943467473, −8.784387501965019, −8.459985009255354, −7.993512612498278, −7.282290929812621, −7.003118205254935, −6.381978511703624, −5.680761125823385, −5.266233162707213, −4.658051506725901, −4.065261727006234, −3.584329366469850, −2.921976669415375, −2.316775058897379, −1.477568618783892, −0.9310379548583432, 0, 0.9310379548583432, 1.477568618783892, 2.316775058897379, 2.921976669415375, 3.584329366469850, 4.065261727006234, 4.658051506725901, 5.266233162707213, 5.680761125823385, 6.381978511703624, 7.003118205254935, 7.282290929812621, 7.993512612498278, 8.459985009255354, 8.784387501965019, 9.466782943467473, 9.924145655805277, 10.53302218808692, 10.99754575696697, 11.44323775900183, 11.87005641639079, 12.26703252273047, 13.01546655799335, 13.33231488925419, 13.76441712219253

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