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Properties

Label 2-230-5.3-c2-0-19
Degree $2$
Conductor $230$
Sign $-0.973 + 0.229i$
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 − i)2-s + (−3.54 − 3.54i)3-s − 2i·4-s + (3.53 − 3.53i)5-s − 7.09·6-s + (9.46 − 9.46i)7-s + (−2 − 2i)8-s + 16.2i·9-s + (−0.00158 − 7.07i)10-s + 7.06·11-s + (−7.09 + 7.09i)12-s + (−6.95 − 6.95i)13-s − 18.9i·14-s + (−25.1 + 0.00563i)15-s − 4·16-s + (−5.69 + 5.69i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−1.18 − 1.18i)3-s − 0.5i·4-s + (0.706 − 0.707i)5-s − 1.18·6-s + (1.35 − 1.35i)7-s + (−0.250 − 0.250i)8-s + 1.80i·9-s + (−0.000158 − 0.707i)10-s + 0.642·11-s + (−0.591 + 0.591i)12-s + (−0.535 − 0.535i)13-s − 1.35i·14-s + (−1.67 + 0.000375i)15-s − 0.250·16-s + (−0.335 + 0.335i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.196830 - 1.68884i\)
\(L(\frac12)\) \(\approx\) \(0.196830 - 1.68884i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + (-3.53 + 3.53i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good3 \( 1 + (3.54 + 3.54i)T + 9iT^{2} \)
7 \( 1 + (-9.46 + 9.46i)T - 49iT^{2} \)
11 \( 1 - 7.06T + 121T^{2} \)
13 \( 1 + (6.95 + 6.95i)T + 169iT^{2} \)
17 \( 1 + (5.69 - 5.69i)T - 289iT^{2} \)
19 \( 1 - 34.4iT - 361T^{2} \)
29 \( 1 - 17.8iT - 841T^{2} \)
31 \( 1 - 44.2T + 961T^{2} \)
37 \( 1 + (0.847 - 0.847i)T - 1.36e3iT^{2} \)
41 \( 1 + 38.8T + 1.68e3T^{2} \)
43 \( 1 + (-20.7 - 20.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (-16.1 + 16.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (-67.0 - 67.0i)T + 2.80e3iT^{2} \)
59 \( 1 + 39.2iT - 3.48e3T^{2} \)
61 \( 1 + 10.2T + 3.72e3T^{2} \)
67 \( 1 + (32.8 - 32.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 50.4T + 5.04e3T^{2} \)
73 \( 1 + (66.5 + 66.5i)T + 5.32e3iT^{2} \)
79 \( 1 - 10.1iT - 6.24e3T^{2} \)
83 \( 1 + (-69.4 - 69.4i)T + 6.88e3iT^{2} \)
89 \( 1 + 54.6iT - 7.92e3T^{2} \)
97 \( 1 + (29.8 - 29.8i)T - 9.40e3iT^{2} \)
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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

−11.83895574247894895104344900485, −10.74300723723471510164480979487, −10.13180270648497457381423450309, −8.318026930731514571313900425594, −7.33490836109372620884537273460, −6.19547352595628049286110862643, −5.25917123680248893903125399366, −4.30953969157230452091423603948, −1.71019030697406711929621946990, −1.02023361275348238366164632189, 2.51945199955167793350318497458, 4.48685726511823684173640422439, 5.11619037026056275527727243139, 6.01704200695727855783694777840, 6.95878211453011558919687401283, 8.750751775515436823933221327269, 9.509451177329511609845943016027, 10.73420782701982999506061362443, 11.68031605824978367193489062139, 11.83289803947901347380792857670

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