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Properties

Label 2-230-115.107-c1-0-2
Degree $2$
Conductor $230$
Sign $0.999 - 0.0255i$
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  + (−0.800 + 0.599i)2-s + (−0.521 − 0.194i)3-s + (0.281 − 0.959i)4-s + (2.23 − 0.0815i)5-s + (0.533 − 0.156i)6-s + (−0.513 + 0.111i)7-s + (0.349 + 0.936i)8-s + (−2.03 − 1.76i)9-s + (−1.73 + 1.40i)10-s + (5.19 − 0.747i)11-s + (−0.333 + 0.445i)12-s + (0.280 − 1.28i)13-s + (0.343 − 0.396i)14-s + (−1.18 − 0.391i)15-s + (−0.841 − 0.540i)16-s + (0.506 − 0.927i)17-s + ⋯
L(s)  = 1  + (−0.566 + 0.423i)2-s + (−0.300 − 0.112i)3-s + (0.140 − 0.479i)4-s + (0.999 − 0.0364i)5-s + (0.217 − 0.0639i)6-s + (−0.193 + 0.0421i)7-s + (0.123 + 0.331i)8-s + (−0.677 − 0.587i)9-s + (−0.550 + 0.444i)10-s + (1.56 − 0.225i)11-s + (−0.0962 + 0.128i)12-s + (0.0776 − 0.357i)13-s + (0.0919 − 0.106i)14-s + (−0.304 − 0.101i)15-s + (−0.210 − 0.135i)16-s + (0.122 − 0.224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.999 - 0.0255i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.999 - 0.0255i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01620 + 0.0129929i\)
\(L(\frac12)\) \(\approx\) \(1.01620 + 0.0129929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.800 - 0.599i)T \)
5 \( 1 + (-2.23 + 0.0815i)T \)
23 \( 1 + (0.244 - 4.78i)T \)
good3 \( 1 + (0.521 + 0.194i)T + (2.26 + 1.96i)T^{2} \)
7 \( 1 + (0.513 - 0.111i)T + (6.36 - 2.90i)T^{2} \)
11 \( 1 + (-5.19 + 0.747i)T + (10.5 - 3.09i)T^{2} \)
13 \( 1 + (-0.280 + 1.28i)T + (-11.8 - 5.40i)T^{2} \)
17 \( 1 + (-0.506 + 0.927i)T + (-9.19 - 14.3i)T^{2} \)
19 \( 1 + (-7.10 - 2.08i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (2.19 + 7.47i)T + (-24.3 + 15.6i)T^{2} \)
31 \( 1 + (0.970 - 2.12i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-0.0455 - 0.636i)T + (-36.6 + 5.26i)T^{2} \)
41 \( 1 + (4.77 + 5.51i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (3.34 - 8.96i)T + (-32.4 - 28.1i)T^{2} \)
47 \( 1 + (8.00 + 8.00i)T + 47iT^{2} \)
53 \( 1 + (0.298 + 1.37i)T + (-48.2 + 22.0i)T^{2} \)
59 \( 1 + (2.55 + 3.98i)T + (-24.5 + 53.6i)T^{2} \)
61 \( 1 + (-8.94 - 4.08i)T + (39.9 + 46.1i)T^{2} \)
67 \( 1 + (2.27 + 3.04i)T + (-18.8 + 64.2i)T^{2} \)
71 \( 1 + (1.06 - 7.41i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (10.9 - 5.95i)T + (39.4 - 61.4i)T^{2} \)
79 \( 1 + (5.14 - 3.30i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (4.31 - 0.308i)T + (82.1 - 11.8i)T^{2} \)
89 \( 1 + (-3.33 - 7.30i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-5.84 - 0.418i)T + (96.0 + 13.8i)T^{2} \)
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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

−11.90908676650946538359875963512, −11.37811832900808860752886359302, −9.819407989604099537674274051965, −9.482333200945434228009126634759, −8.412343367822047207806591097182, −7.00736936196658793486656312590, −6.11380292891847649855618250752, −5.39288657568753062708047490636, −3.35864099835512334166820209544, −1.32254465836713161480874291761, 1.60080432458950548837993333137, 3.18436522257024385599893428606, 4.88776543801370634455200858133, 6.16259821485370141344956472180, 7.11030960059566481239349473647, 8.639039318663093930163937719821, 9.367959908966703784488337419877, 10.19423703184750541547170172184, 11.23479254195187633982257925879, 11.92236955417327886109494130652

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