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Properties

Label 2-230-5.4-c1-0-8
Degree $2$
Conductor $230$
Sign $0.172 + 0.984i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + 1.44i·3-s − 4-s + (−0.386 − 2.20i)5-s + 1.44·6-s − 3.25i·7-s + i·8-s + 0.918·9-s + (−2.20 + 0.386i)10-s + 1.32·11-s − 1.44i·12-s − 1.25i·13-s − 3.25·14-s + (3.17 − 0.557i)15-s + 16-s − 4.62i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.832i·3-s − 0.5·4-s + (−0.172 − 0.984i)5-s + 0.588·6-s − 1.23i·7-s + 0.353i·8-s + 0.306·9-s + (−0.696 + 0.122i)10-s + 0.400·11-s − 0.416i·12-s − 0.349i·13-s − 0.870·14-s + (0.820 − 0.143i)15-s + 0.250·16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.172 + 0.984i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.172 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886013 - 0.744139i\)
\(L(\frac12)\) \(\approx\) \(0.886013 - 0.744139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (0.386 + 2.20i)T \)
23 \( 1 - iT \)
good3 \( 1 - 1.44iT - 3T^{2} \)
7 \( 1 + 3.25iT - 7T^{2} \)
11 \( 1 - 1.32T + 11T^{2} \)
13 \( 1 + 1.25iT - 13T^{2} \)
17 \( 1 + 4.62iT - 17T^{2} \)
19 \( 1 - 3.37T + 19T^{2} \)
29 \( 1 + 4.29T + 29T^{2} \)
31 \( 1 - 2.37T + 31T^{2} \)
37 \( 1 - 5.74iT - 37T^{2} \)
41 \( 1 + 5.13T + 41T^{2} \)
43 \( 1 - 10.4iT - 43T^{2} \)
47 \( 1 - 6.06iT - 47T^{2} \)
53 \( 1 + 11.1iT - 53T^{2} \)
59 \( 1 - 2.72T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 - 6.60iT - 67T^{2} \)
71 \( 1 - 0.265T + 71T^{2} \)
73 \( 1 - 5.26iT - 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + 2.02iT - 83T^{2} \)
89 \( 1 + 16.1T + 89T^{2} \)
97 \( 1 - 8.62iT - 97T^{2} \)
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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

−11.80653373934240017421370362723, −11.05430327256961847856108485483, −9.818430125170247779592203858178, −9.600571049686466748679035761791, −8.244064641503819332909243833441, −7.12365876883511987211192060107, −5.20447214403663964009011231240, −4.40406385064863517662828134551, −3.46710513104762874453542438086, −1.09483250628347069675827836250, 2.10838443195247367694251616977, 3.80342713855164344277217654788, 5.57943991841161642436734188439, 6.48259210456133866145263604537, 7.25881588684253249655447475491, 8.252386437065452427150762996127, 9.293420438080555586680006193562, 10.45338045945580285838423529513, 11.75784811492849671605186752797, 12.38494624893549804659837125433

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