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Properties

Label 2-230-115.114-c4-0-6
Degree $2$
Conductor $230$
Sign $0.728 - 0.685i$
Analytic cond. $23.7750$
Root an. cond. $4.87597$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82i·2-s − 9.09i·3-s − 8.00·4-s + (21.5 − 12.6i)5-s − 25.7·6-s − 94.0·7-s + 22.6i·8-s − 1.77·9-s + (−35.7 − 60.9i)10-s + 189. i·11-s + 72.7i·12-s − 27.4i·13-s + 266. i·14-s + (−115. − 196. i)15-s + 64.0·16-s − 179.·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.01i·3-s − 0.500·4-s + (0.862 − 0.505i)5-s − 0.714·6-s − 1.91·7-s + 0.353i·8-s − 0.0219·9-s + (−0.357 − 0.609i)10-s + 1.56i·11-s + 0.505i·12-s − 0.162i·13-s + 1.35i·14-s + (−0.511 − 0.871i)15-s + 0.250·16-s − 0.622·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.728 - 0.685i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.728 - 0.685i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4828490381\)
\(L(\frac12)\) \(\approx\) \(0.4828490381\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
5 \( 1 + (-21.5 + 12.6i)T \)
23 \( 1 + (-117. - 515. i)T \)
good3 \( 1 + 9.09iT - 81T^{2} \)
7 \( 1 + 94.0T + 2.40e3T^{2} \)
11 \( 1 - 189. iT - 1.46e4T^{2} \)
13 \( 1 + 27.4iT - 2.85e4T^{2} \)
17 \( 1 + 179.T + 8.35e4T^{2} \)
19 \( 1 - 320. iT - 1.30e5T^{2} \)
29 \( 1 + 801.T + 7.07e5T^{2} \)
31 \( 1 + 714.T + 9.23e5T^{2} \)
37 \( 1 + 1.76e3T + 1.87e6T^{2} \)
41 \( 1 - 990.T + 2.82e6T^{2} \)
43 \( 1 - 1.57e3T + 3.41e6T^{2} \)
47 \( 1 - 992. iT - 4.87e6T^{2} \)
53 \( 1 - 1.10e3T + 7.89e6T^{2} \)
59 \( 1 + 4.65e3T + 1.21e7T^{2} \)
61 \( 1 - 3.79e3iT - 1.38e7T^{2} \)
67 \( 1 - 4.64e3T + 2.01e7T^{2} \)
71 \( 1 + 3.89e3T + 2.54e7T^{2} \)
73 \( 1 + 4.78e3iT - 2.83e7T^{2} \)
79 \( 1 - 7.38e3iT - 3.89e7T^{2} \)
83 \( 1 - 940.T + 4.74e7T^{2} \)
89 \( 1 - 1.03e4iT - 6.27e7T^{2} \)
97 \( 1 + 3.38e3T + 8.85e7T^{2} \)
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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

−12.23470423591827606149435592537, −10.48462336697028148089693651687, −9.658053647497763272210853001344, −9.180685382441142101504330731804, −7.49212571162888121020727960217, −6.62761182739870774814897709457, −5.57730644156438580840631611916, −3.96270364626962556550058071102, −2.43914303852689811716297137919, −1.44269871383307840134700672600, 0.16021021275172284095931696423, 2.92129353564028863278667123992, 3.84807806611381934344031359974, 5.41369615379365095071168742142, 6.30912362575136273719460319824, 7.01645422905718604920454261197, 9.008754248559242529548674485881, 9.251154099240834660802453292536, 10.33424351399186443465621676916, 10.96550037599588959173600954081

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