Properties

Label 2-229320-1.1-c1-0-24
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 + 2·11-s − 13-s − 6·17-s − 6·19-s − 6·23-s + 25-s + 6·29-s + 10·31-s − 6·37-s + 10·41-s + 2·43-s + 10·53-s + 2·55-s − 14·59-s − 2·61-s − 65-s + 16·67-s − 6·71-s + 2·73-s + 4·79-s + 4·83-s − 6·85-s + 2·89-s − 6·95-s + 14·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.603·11-s − 0.277·13-s − 1.45·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.79·31-s − 0.986·37-s + 1.56·41-s + 0.304·43-s + 1.37·53-s + 0.269·55-s − 1.82·59-s − 0.256·61-s − 0.124·65-s + 1.95·67-s − 0.712·71-s + 0.234·73-s + 0.450·79-s + 0.439·83-s − 0.650·85-s + 0.211·89-s − 0.615·95-s + 1.42·97-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

\(L(1)\) \(\approx\) \(2.171791076\)
\(L(\frac12)\) \(\approx\) \(2.171791076\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−12.84673514044159, −12.47110264400151, −12.07544348467486, −11.56304732560501, −11.05374010003417, −10.48707356066434, −10.26408491627312, −9.672725000065886, −9.136280622144277, −8.717697263176348, −8.335499362922461, −7.808056053228486, −7.124656528517723, −6.589921689423477, −6.239877401839111, −5.990844324129051, −5.083987623763448, −4.564944614041985, −4.252579239095065, −3.710582241398352, −2.843553039019467, −2.291255534276616, −2.019472564361499, −1.130605072488608, −0.4238387050987667, 0.4238387050987667, 1.130605072488608, 2.019472564361499, 2.291255534276616, 2.843553039019467, 3.710582241398352, 4.252579239095065, 4.564944614041985, 5.083987623763448, 5.990844324129051, 6.239877401839111, 6.589921689423477, 7.124656528517723, 7.808056053228486, 8.335499362922461, 8.717697263176348, 9.136280622144277, 9.672725000065886, 10.26408491627312, 10.48707356066434, 11.05374010003417, 11.56304732560501, 12.07544348467486, 12.47110264400151, 12.84673514044159

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