Properties

Label 2-229320-1.1-c1-0-96
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 − 4·11-s − 13-s + 3·17-s − 6·19-s + 9·23-s + 25-s − 6·31-s − 6·37-s − 5·41-s + 6·43-s + 3·53-s − 4·55-s + 13·59-s + 5·61-s − 65-s + 5·67-s + 71-s + 7·73-s − 6·83-s + 3·85-s − 14·89-s − 6·95-s + 97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s − 0.277·13-s + 0.727·17-s − 1.37·19-s + 1.87·23-s + 1/5·25-s − 1.07·31-s − 0.986·37-s − 0.780·41-s + 0.914·43-s + 0.412·53-s − 0.539·55-s + 1.69·59-s + 0.640·61-s − 0.124·65-s + 0.610·67-s + 0.118·71-s + 0.819·73-s − 0.658·83-s + 0.325·85-s − 1.48·89-s − 0.615·95-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.15545144795795, −12.80040169593141, −12.37450694255986, −11.82838041237620, −11.11018542608526, −10.79021182528222, −10.50080998552088, −9.868913684597202, −9.548421487457864, −8.849427337025481, −8.497189669918066, −8.094268525094860, −7.337156580479957, −7.021707496569215, −6.630889377541133, −5.750142172884948, −5.495023379478554, −5.034336471271853, −4.538391352342511, −3.747908516963674, −3.313477188817800, −2.540380949504762, −2.290855756991801, −1.515591824715722, −0.7678692564297166, 0, 0.7678692564297166, 1.515591824715722, 2.290855756991801, 2.540380949504762, 3.313477188817800, 3.747908516963674, 4.538391352342511, 5.034336471271853, 5.495023379478554, 5.750142172884948, 6.630889377541133, 7.021707496569215, 7.337156580479957, 8.094268525094860, 8.497189669918066, 8.849427337025481, 9.548421487457864, 9.868913684597202, 10.50080998552088, 10.79021182528222, 11.11018542608526, 11.82838041237620, 12.37450694255986, 12.80040169593141, 13.15545144795795

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