Properties

Label 2-230-23.22-c2-0-10
Degree $2$
Conductor $230$
Sign $-0.729 + 0.684i$
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  − 1.41·2-s − 0.278·3-s + 2.00·4-s + 2.23i·5-s + 0.393·6-s − 8.51i·7-s − 2.82·8-s − 8.92·9-s − 3.16i·10-s + 7.57i·11-s − 0.557·12-s − 2.64·13-s + 12.0i·14-s − 0.622i·15-s + 4.00·16-s − 7.56i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0928·3-s + 0.500·4-s + 0.447i·5-s + 0.0656·6-s − 1.21i·7-s − 0.353·8-s − 0.991·9-s − 0.316i·10-s + 0.688i·11-s − 0.0464·12-s − 0.203·13-s + 0.859i·14-s − 0.0415i·15-s + 0.250·16-s − 0.444i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.729 + 0.684i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ -0.729 + 0.684i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.152586 - 0.385658i\)
\(L(\frac12)\) \(\approx\) \(0.152586 - 0.385658i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
5 \( 1 - 2.23iT \)
23 \( 1 + (15.7 + 16.7i)T \)
good3 \( 1 + 0.278T + 9T^{2} \)
7 \( 1 + 8.51iT - 49T^{2} \)
11 \( 1 - 7.57iT - 121T^{2} \)
13 \( 1 + 2.64T + 169T^{2} \)
17 \( 1 + 7.56iT - 289T^{2} \)
19 \( 1 + 24.2iT - 361T^{2} \)
29 \( 1 + 31.8T + 841T^{2} \)
31 \( 1 + 56.5T + 961T^{2} \)
37 \( 1 + 39.9iT - 1.36e3T^{2} \)
41 \( 1 + 42.5T + 1.68e3T^{2} \)
43 \( 1 + 20.5iT - 1.84e3T^{2} \)
47 \( 1 - 84.3T + 2.20e3T^{2} \)
53 \( 1 + 11.9iT - 2.80e3T^{2} \)
59 \( 1 - 67.6T + 3.48e3T^{2} \)
61 \( 1 - 35.1iT - 3.72e3T^{2} \)
67 \( 1 + 44.0iT - 4.48e3T^{2} \)
71 \( 1 - 8.86T + 5.04e3T^{2} \)
73 \( 1 + 87.4T + 5.32e3T^{2} \)
79 \( 1 - 154. iT - 6.24e3T^{2} \)
83 \( 1 - 141. iT - 6.88e3T^{2} \)
89 \( 1 - 63.7iT - 7.92e3T^{2} \)
97 \( 1 + 143. iT - 9.40e3T^{2} \)
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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

−11.23597153345514675507233931823, −10.73215527042834028055062008066, −9.704694321509841653678691143198, −8.744998396213044015027086475045, −7.41714908379398568404599239551, −6.94829758791045824328871945740, −5.47873088920079117604371873864, −3.90177262250377595157959311900, −2.35800676987130341050682166084, −0.26582247632232933160156770569, 1.92328080344107621391033727706, 3.44437267778733516590706150213, 5.53653996597129220980600096876, 5.96519408649310293199667554094, 7.68835552015821492038969511663, 8.604751700610843117387991525254, 9.163125438344200511742793511627, 10.34830455333530776618073894837, 11.52110560124270692466026843931, 12.01189379255322996791893581491

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