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Properties

Label 2-230-5.4-c1-0-3
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

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

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