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Properties

Label 2-234-13.12-c7-0-21
Degree $2$
Conductor $234$
Sign $0.999 + 0.0302i$
Analytic cond. $73.0980$
Root an. cond. $8.54974$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8i·2-s − 64·4-s − 197. i·5-s − 828. i·7-s − 512i·8-s + 1.57e3·10-s + 2.52e3i·11-s + (−239. + 7.91e3i)13-s + 6.62e3·14-s + 4.09e3·16-s − 1.67e4·17-s + 4.51e4i·19-s + 1.26e4i·20-s − 2.02e4·22-s − 5.36e4·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.706i·5-s − 0.912i·7-s − 0.353i·8-s + 0.499·10-s + 0.572i·11-s + (−0.0302 + 0.999i)13-s + 0.645·14-s + 0.250·16-s − 0.826·17-s + 1.51i·19-s + 0.353i·20-s − 0.404·22-s − 0.920·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0302i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.999 + 0.0302i$
Analytic conductor: \(73.0980\)
Root analytic conductor: \(8.54974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :7/2),\ 0.999 + 0.0302i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.697419626\)
\(L(\frac12)\) \(\approx\) \(1.697419626\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 8iT \)
3 \( 1 \)
13 \( 1 + (239. - 7.91e3i)T \)
good5 \( 1 + 197. iT - 7.81e4T^{2} \)
7 \( 1 + 828. iT - 8.23e5T^{2} \)
11 \( 1 - 2.52e3iT - 1.94e7T^{2} \)
17 \( 1 + 1.67e4T + 4.10e8T^{2} \)
19 \( 1 - 4.51e4iT - 8.93e8T^{2} \)
23 \( 1 + 5.36e4T + 3.40e9T^{2} \)
29 \( 1 - 1.57e5T + 1.72e10T^{2} \)
31 \( 1 + 1.28e5iT - 2.75e10T^{2} \)
37 \( 1 + 9.56e4iT - 9.49e10T^{2} \)
41 \( 1 + 8.44e5iT - 1.94e11T^{2} \)
43 \( 1 - 9.17e5T + 2.71e11T^{2} \)
47 \( 1 - 5.97e5iT - 5.06e11T^{2} \)
53 \( 1 + 5.76e5T + 1.17e12T^{2} \)
59 \( 1 + 1.63e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.55e6T + 3.14e12T^{2} \)
67 \( 1 + 8.24e5iT - 6.06e12T^{2} \)
71 \( 1 + 5.63e6iT - 9.09e12T^{2} \)
73 \( 1 - 2.72e6iT - 1.10e13T^{2} \)
79 \( 1 + 3.75e6T + 1.92e13T^{2} \)
83 \( 1 - 1.52e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.74e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.06e7iT - 8.07e13T^{2} \)
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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

−10.75123846585270601298386268427, −9.780721900121122604093587968789, −8.864642111487069548427301716278, −7.86220763891860807578057217838, −6.95667807929009909613781371402, −5.92115544115684914523009696089, −4.56029033273465853590982664543, −3.98549315011069810658788984988, −1.89632325012941739919426921886, −0.57037342486153059289766469848, 0.793751423629071173132837265747, 2.49456389513205205612119693509, 3.02481915747946150820317717286, 4.57926514130120446325305835511, 5.74999131417956477282142670908, 6.82983587027317691980176951809, 8.259857707877860120937311554939, 9.001612413426898609134828398437, 10.16734257403743022827289175157, 10.96785367649612617708674962736

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