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Properties

Label 2-92400-1.1-c1-0-139
Degree $2$
Conductor $92400$
Sign $-1$
Analytic cond. $737.817$
Root an. cond. $27.1628$
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  + 3-s − 7-s + 9-s − 11-s − 2·13-s − 2·17-s + 6·19-s − 21-s + 27-s − 6·29-s − 4·31-s − 33-s − 2·39-s + 10·41-s − 8·43-s − 8·47-s + 49-s − 2·51-s + 14·53-s + 6·57-s − 2·59-s − 63-s + 14·67-s − 4·71-s − 2·73-s + 77-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.485·17-s + 1.37·19-s − 0.218·21-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.174·33-s − 0.320·39-s + 1.56·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.280·51-s + 1.92·53-s + 0.794·57-s − 0.260·59-s − 0.125·63-s + 1.71·67-s − 0.474·71-s − 0.234·73-s + 0.113·77-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(737.817\)
Root analytic conductor: \(27.1628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.02053117237900, −13.59148492107248, −13.12619108822522, −12.73867156088759, −12.20352721995989, −11.53768512742381, −11.26082670049633, −10.50196743713333, −10.07543090022352, −9.416868497777899, −9.314003257377118, −8.628169780844500, −7.949945313931842, −7.605666809512471, −7.035684282728942, −6.635148490133331, −5.791274116101189, −5.339604215060880, −4.828428958566246, −4.018668342304268, −3.588908597595963, −2.950926292276599, −2.378816451245344, −1.768618539223000, −0.8937262179972306, 0, 0.8937262179972306, 1.768618539223000, 2.378816451245344, 2.950926292276599, 3.588908597595963, 4.018668342304268, 4.828428958566246, 5.339604215060880, 5.791274116101189, 6.635148490133331, 7.035684282728942, 7.605666809512471, 7.949945313931842, 8.628169780844500, 9.314003257377118, 9.416868497777899, 10.07543090022352, 10.50196743713333, 11.26082670049633, 11.53768512742381, 12.20352721995989, 12.73867156088759, 13.12619108822522, 13.59148492107248, 14.02053117237900

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