Properties

Label 2-229320-1.1-c1-0-15
Degree $2$
Conductor $229320$
Sign $1$
Analytic cond. $1831.12$
Root an. cond. $42.7916$
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 − 5·11-s − 13-s − 7·17-s + 6·19-s + 3·23-s + 25-s − 6·29-s + 4·31-s + 7·37-s − 3·41-s + 10·47-s − 6·53-s + 5·55-s + 9·59-s − 2·61-s + 65-s − 9·67-s + 16·71-s − 73-s + 3·79-s − 16·83-s + 7·85-s − 5·89-s − 6·95-s + 17·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.50·11-s − 0.277·13-s − 1.69·17-s + 1.37·19-s + 0.625·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 1.15·37-s − 0.468·41-s + 1.45·47-s − 0.824·53-s + 0.674·55-s + 1.17·59-s − 0.256·61-s + 0.124·65-s − 1.09·67-s + 1.89·71-s − 0.117·73-s + 0.337·79-s − 1.75·83-s + 0.759·85-s − 0.529·89-s − 0.615·95-s + 1.72·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(229320\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1831.12\)
Root analytic conductor: \(42.7916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{229320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.290688629\)
\(L(\frac12)\) \(\approx\) \(1.290688629\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
good11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−12.92628463363925, −12.63924712388265, −11.88759541976361, −11.53379524960283, −10.99675700805495, −10.81288039614097, −10.11238004150533, −9.687539906152472, −9.189268648388718, −8.693140178437397, −8.155856479568762, −7.732852856915023, −7.204130098362933, −6.975980695491218, −6.170903599648502, −5.663623899823959, −5.153307476661520, −4.678787401818348, −4.252972494276364, −3.512902356600709, −2.947498191814915, −2.477377184402218, −1.963376288477511, −0.9902328355465076, −0.3566565163938399, 0.3566565163938399, 0.9902328355465076, 1.963376288477511, 2.477377184402218, 2.947498191814915, 3.512902356600709, 4.252972494276364, 4.678787401818348, 5.153307476661520, 5.663623899823959, 6.170903599648502, 6.975980695491218, 7.204130098362933, 7.732852856915023, 8.155856479568762, 8.693140178437397, 9.189268648388718, 9.687539906152472, 10.11238004150533, 10.81288039614097, 10.99675700805495, 11.53379524960283, 11.88759541976361, 12.63924712388265, 12.92628463363925

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