Properties

Label 2-229320-1.1-c1-0-101
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s + 13-s + 7·17-s − 2·19-s − 3·23-s + 25-s − 6·29-s − 8·31-s − 37-s + 9·41-s + 2·43-s + 8·47-s + 6·53-s − 55-s + 3·59-s − 14·61-s − 65-s + 15·67-s + 8·71-s + 73-s − 79-s + 8·83-s − 7·85-s + 15·89-s + 2·95-s + 3·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s + 0.277·13-s + 1.69·17-s − 0.458·19-s − 0.625·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.164·37-s + 1.40·41-s + 0.304·43-s + 1.16·47-s + 0.824·53-s − 0.134·55-s + 0.390·59-s − 1.79·61-s − 0.124·65-s + 1.83·67-s + 0.949·71-s + 0.117·73-s − 0.112·79-s + 0.878·83-s − 0.759·85-s + 1.58·89-s + 0.205·95-s + 0.304·97-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: \(1\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 - T \)
good11 \( 1 - T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−13.12178982541576, −12.52654245958536, −12.31843029600230, −11.84911259672785, −11.30023323762013, −10.74880833108262, −10.58403540084691, −9.847101778464591, −9.326333073639923, −9.098489981120598, −8.405172523071667, −7.872003157237617, −7.524245098291879, −7.190685811872473, −6.341594059467672, −6.047712845544009, −5.341568263839371, −5.161027546736063, −4.171678296913767, −3.754619759603122, −3.596225104318705, −2.668404384566366, −2.159635281559504, −1.393304596802603, −0.8290221597073889, 0, 0.8290221597073889, 1.393304596802603, 2.159635281559504, 2.668404384566366, 3.596225104318705, 3.754619759603122, 4.171678296913767, 5.161027546736063, 5.341568263839371, 6.047712845544009, 6.341594059467672, 7.190685811872473, 7.524245098291879, 7.872003157237617, 8.405172523071667, 9.098489981120598, 9.326333073639923, 9.847101778464591, 10.58403540084691, 10.74880833108262, 11.30023323762013, 11.84911259672785, 12.31843029600230, 12.52654245958536, 13.12178982541576

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