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Properties

Label 2-230-115.114-c2-0-12
Degree $2$
Conductor $230$
Sign $0.892 - 0.451i$
Analytic cond. $6.26704$
Root an. cond. $2.50340$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 1.47i·3-s − 2.00·4-s + (4.75 + 1.55i)5-s + 2.08·6-s − 0.788·7-s − 2.82i·8-s + 6.82·9-s + (−2.20 + 6.71i)10-s − 20.2i·11-s + 2.95i·12-s + 19.5i·13-s − 1.11i·14-s + (2.29 − 7.01i)15-s + 4.00·16-s + 15.2·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.492i·3-s − 0.500·4-s + (0.950 + 0.311i)5-s + 0.347·6-s − 0.112·7-s − 0.353i·8-s + 0.757·9-s + (−0.220 + 0.671i)10-s − 1.84i·11-s + 0.246i·12-s + 1.50i·13-s − 0.0796i·14-s + (0.153 − 0.467i)15-s + 0.250·16-s + 0.895·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.892 - 0.451i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.892 - 0.451i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.78609 + 0.425713i\)
\(L(\frac12)\) \(\approx\) \(1.78609 + 0.425713i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
5 \( 1 + (-4.75 - 1.55i)T \)
23 \( 1 + (-16.2 - 16.2i)T \)
good3 \( 1 + 1.47iT - 9T^{2} \)
7 \( 1 + 0.788T + 49T^{2} \)
11 \( 1 + 20.2iT - 121T^{2} \)
13 \( 1 - 19.5iT - 169T^{2} \)
17 \( 1 - 15.2T + 289T^{2} \)
19 \( 1 + 2.22iT - 361T^{2} \)
29 \( 1 - 33.2T + 841T^{2} \)
31 \( 1 - 15.3T + 961T^{2} \)
37 \( 1 - 5.15T + 1.36e3T^{2} \)
41 \( 1 + 63.8T + 1.68e3T^{2} \)
43 \( 1 + 16.4T + 1.84e3T^{2} \)
47 \( 1 - 20.8iT - 2.20e3T^{2} \)
53 \( 1 + 46.8T + 2.80e3T^{2} \)
59 \( 1 + 36.3T + 3.48e3T^{2} \)
61 \( 1 + 89.3iT - 3.72e3T^{2} \)
67 \( 1 + 64.3T + 4.48e3T^{2} \)
71 \( 1 - 28.5T + 5.04e3T^{2} \)
73 \( 1 + 83.5iT - 5.32e3T^{2} \)
79 \( 1 - 27.5iT - 6.24e3T^{2} \)
83 \( 1 - 55.6T + 6.88e3T^{2} \)
89 \( 1 + 12.2iT - 7.92e3T^{2} \)
97 \( 1 - 43.7T + 9.40e3T^{2} \)
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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

−12.18428262920293914456918461076, −11.06399159909780775728288948958, −9.907342384582623990013305860478, −9.066144848964171325845908650998, −7.987464484449649430480791513435, −6.71582252741788908612595392246, −6.24347410335687242125806794849, −4.96895202773175158468991826726, −3.28082461845888210717579458661, −1.36326846780316430594183354036, 1.41773983167975550906946951422, 2.92136625810579133794440125878, 4.53684457755648195548906059790, 5.27214763154074078000104920127, 6.82014747670457617919522778070, 8.145949642475587913374657910146, 9.455286741155408414131800180992, 10.15607741799990315608960117732, 10.42390835557323566702462268726, 12.20436394053775684648940979018

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