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Properties

Label 2-230-115.103-c1-0-10
Degree $2$
Conductor $230$
Sign $-0.475 + 0.879i$
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  + (0.212 − 0.977i)2-s + (0.521 − 0.390i)3-s + (−0.909 − 0.415i)4-s + (−0.172 − 2.22i)5-s + (−0.270 − 0.593i)6-s + (0.196 − 2.75i)7-s + (−0.599 + 0.800i)8-s + (−0.725 + 2.47i)9-s + (−2.21 − 0.305i)10-s + (−0.171 + 0.266i)11-s + (−0.637 + 0.138i)12-s + (−0.610 + 0.0436i)13-s + (−2.64 − 0.777i)14-s + (−0.960 − 1.09i)15-s + (0.654 + 0.755i)16-s + (0.930 − 2.49i)17-s + ⋯
L(s)  = 1  + (0.150 − 0.690i)2-s + (0.301 − 0.225i)3-s + (−0.454 − 0.207i)4-s + (−0.0770 − 0.997i)5-s + (−0.110 − 0.242i)6-s + (0.0744 − 1.04i)7-s + (−0.211 + 0.283i)8-s + (−0.241 + 0.823i)9-s + (−0.700 − 0.0966i)10-s + (−0.0516 + 0.0803i)11-s + (−0.183 + 0.0400i)12-s + (−0.169 + 0.0121i)13-s + (−0.707 − 0.207i)14-s + (−0.248 − 0.283i)15-s + (0.163 + 0.188i)16-s + (0.225 − 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.475 + 0.879i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ -0.475 + 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.671544 - 1.12617i\)
\(L(\frac12)\) \(\approx\) \(0.671544 - 1.12617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.212 + 0.977i)T \)
5 \( 1 + (0.172 + 2.22i)T \)
23 \( 1 + (-4.61 + 1.29i)T \)
good3 \( 1 + (-0.521 + 0.390i)T + (0.845 - 2.87i)T^{2} \)
7 \( 1 + (-0.196 + 2.75i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (0.171 - 0.266i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (0.610 - 0.0436i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (-0.930 + 2.49i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (-1.33 + 2.93i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-5.55 + 2.53i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (-1.29 - 8.98i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-1.15 - 2.11i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (7.62 - 2.23i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-6.26 - 8.36i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (-4.91 + 4.91i)T - 47iT^{2} \)
53 \( 1 + (-5.77 - 0.413i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (3.03 + 2.63i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (10.7 - 1.53i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (-7.50 - 1.63i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (-7.02 + 4.51i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (9.02 - 3.36i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (4.91 - 5.66i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-0.897 + 0.489i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (0.532 - 3.70i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (10.5 + 5.77i)T + (52.4 + 81.6i)T^{2} \)
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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

−11.95119254556998191044570045129, −10.96007431779853289644849553433, −10.07473209260523807066217875127, −8.978971246152799491973843289470, −8.082770249340690068115995390522, −7.02427712105533526002497829696, −5.17465628238516634868993785624, −4.47424896918792066822554986245, −2.87788146515713751801312679494, −1.12686602365103184088983112492, 2.74171319137302312466030036132, 3.88000249653384774328248015534, 5.56582404674759620677381274479, 6.34736634139802982341450855066, 7.49082257603585676676728169366, 8.577324733686835235928701904712, 9.422697380093481151989980400935, 10.47922548215909852676457618972, 11.74233025121933472546941414860, 12.43779542524574168928950278287

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