Properties

Label 2-229320-1.1-c1-0-63
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 − 6·11-s + 13-s − 2·17-s − 2·19-s + 2·23-s + 25-s − 10·29-s + 6·31-s + 6·37-s − 2·41-s + 10·43-s − 14·53-s + 6·55-s − 2·59-s − 2·61-s − 65-s + 8·67-s + 10·71-s − 2·73-s + 4·79-s − 4·83-s + 2·85-s − 2·89-s + 2·95-s − 14·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.80·11-s + 0.277·13-s − 0.485·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s − 1.85·29-s + 1.07·31-s + 0.986·37-s − 0.312·41-s + 1.52·43-s − 1.92·53-s + 0.809·55-s − 0.260·59-s − 0.256·61-s − 0.124·65-s + 0.977·67-s + 1.18·71-s − 0.234·73-s + 0.450·79-s − 0.439·83-s + 0.216·85-s − 0.211·89-s + 0.205·95-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: \(1\)
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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
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

−13.01332449598237, −12.76765645534518, −12.45065981124848, −11.64194218263865, −11.20604545622991, −10.86632410431658, −10.56220294788649, −9.914532475873858, −9.326371773069019, −9.093953448497357, −8.172347379715470, −8.003446983319758, −7.719621982624651, −6.930560809949510, −6.615365257533771, −5.822258703748175, −5.517724568401044, −4.920499683906113, −4.391227221377917, −3.949932785924468, −3.198113214716254, −2.711723017825653, −2.251992729069150, −1.508600863583103, −0.6164927312815640, 0, 0.6164927312815640, 1.508600863583103, 2.251992729069150, 2.711723017825653, 3.198113214716254, 3.949932785924468, 4.391227221377917, 4.920499683906113, 5.517724568401044, 5.822258703748175, 6.615365257533771, 6.930560809949510, 7.719621982624651, 8.003446983319758, 8.172347379715470, 9.093953448497357, 9.326371773069019, 9.914532475873858, 10.56220294788649, 10.86632410431658, 11.20604545622991, 11.64194218263865, 12.45065981124848, 12.76765645534518, 13.01332449598237

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