Properties

Label 2-229320-1.1-c1-0-42
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 + 6·11-s + 13-s + 2·17-s − 4·19-s + 6·23-s + 25-s − 8·29-s + 4·31-s + 2·37-s + 2·41-s − 4·43-s + 8·47-s + 12·53-s − 6·55-s + 8·59-s − 14·61-s − 65-s − 12·67-s − 10·71-s + 10·73-s − 16·79-s − 4·83-s − 2·85-s + 18·89-s + 4·95-s + 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.80·11-s + 0.277·13-s + 0.485·17-s − 0.917·19-s + 1.25·23-s + 1/5·25-s − 1.48·29-s + 0.718·31-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 1.16·47-s + 1.64·53-s − 0.809·55-s + 1.04·59-s − 1.79·61-s − 0.124·65-s − 1.46·67-s − 1.18·71-s + 1.17·73-s − 1.80·79-s − 0.439·83-s − 0.216·85-s + 1.90·89-s + 0.410·95-s + 0.203·97-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.867031241\)
\(L(\frac12)\) \(\approx\) \(2.867031241\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−12.97178133241038, −12.34792781006749, −11.94418496043272, −11.63284879632057, −11.13000570399239, −10.66877698904119, −10.20966233893103, −9.538967231362387, −9.102914717220489, −8.763170961524154, −8.403178916552360, −7.526047392639961, −7.318957133427196, −6.767322973473566, −6.196546071340746, −5.874461172210526, −5.185923569043000, −4.491958049074854, −4.114763726554664, −3.674827779915051, −3.115199448770507, −2.446492087082706, −1.651547872512809, −1.173091763170080, −0.5118856838145261, 0.5118856838145261, 1.173091763170080, 1.651547872512809, 2.446492087082706, 3.115199448770507, 3.674827779915051, 4.114763726554664, 4.491958049074854, 5.185923569043000, 5.874461172210526, 6.196546071340746, 6.767322973473566, 7.318957133427196, 7.526047392639961, 8.403178916552360, 8.763170961524154, 9.102914717220489, 9.538967231362387, 10.20966233893103, 10.66877698904119, 11.13000570399239, 11.63284879632057, 11.94418496043272, 12.34792781006749, 12.97178133241038

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