Properties

Label 2-230-115.29-c1-0-0
Degree $2$
Conductor $230$
Sign $0.542 - 0.840i$
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  + (−0.909 − 0.415i)2-s + (0.0359 − 0.122i)3-s + (0.654 + 0.755i)4-s + (−1.96 + 1.06i)5-s + (−0.0836 + 0.0965i)6-s + (−0.277 − 0.0399i)7-s + (−0.281 − 0.959i)8-s + (2.51 + 1.61i)9-s + (2.23 − 0.147i)10-s + (2.07 + 4.53i)11-s + (0.116 − 0.0530i)12-s + (−3.69 + 0.531i)13-s + (0.236 + 0.151i)14-s + (0.0591 + 0.279i)15-s + (−0.142 + 0.989i)16-s + (5.80 + 5.02i)17-s + ⋯
L(s)  = 1  + (−0.643 − 0.293i)2-s + (0.0207 − 0.0707i)3-s + (0.327 + 0.377i)4-s + (−0.880 + 0.474i)5-s + (−0.0341 + 0.0394i)6-s + (−0.105 − 0.0151i)7-s + (−0.0996 − 0.339i)8-s + (0.836 + 0.537i)9-s + (0.705 − 0.0465i)10-s + (0.624 + 1.36i)11-s + (0.0335 − 0.0153i)12-s + (−1.02 + 0.147i)13-s + (0.0631 + 0.0405i)14-s + (0.0152 + 0.0721i)15-s + (−0.0355 + 0.247i)16-s + (1.40 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.542 - 0.840i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.542 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.671105 + 0.365685i\)
\(L(\frac12)\) \(\approx\) \(0.671105 + 0.365685i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 + 0.415i)T \)
5 \( 1 + (1.96 - 1.06i)T \)
23 \( 1 + (-0.919 - 4.70i)T \)
good3 \( 1 + (-0.0359 + 0.122i)T + (-2.52 - 1.62i)T^{2} \)
7 \( 1 + (0.277 + 0.0399i)T + (6.71 + 1.97i)T^{2} \)
11 \( 1 + (-2.07 - 4.53i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (3.69 - 0.531i)T + (12.4 - 3.66i)T^{2} \)
17 \( 1 + (-5.80 - 5.02i)T + (2.41 + 16.8i)T^{2} \)
19 \( 1 + (2.73 + 3.15i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (1.53 - 1.77i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-3.18 + 0.936i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (4.55 - 7.09i)T + (-15.3 - 33.6i)T^{2} \)
41 \( 1 + (-1.35 + 0.868i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-2.28 + 7.78i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 + 5.98iT - 47T^{2} \)
53 \( 1 + (10.5 + 1.51i)T + (50.8 + 14.9i)T^{2} \)
59 \( 1 + (1.46 + 10.1i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (-10.4 + 3.05i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-8.30 - 3.79i)T + (43.8 + 50.6i)T^{2} \)
71 \( 1 + (1.57 - 3.44i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (7.70 - 6.67i)T + (10.3 - 72.2i)T^{2} \)
79 \( 1 + (-0.687 - 4.77i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-6.83 + 10.6i)T + (-34.4 - 75.4i)T^{2} \)
89 \( 1 + (5.36 + 1.57i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (-2.46 - 3.83i)T + (-40.2 + 88.2i)T^{2} \)
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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

−12.28878604287971544933409581540, −11.39063418955632590538513593245, −10.21285113133642719784524795262, −9.771693887779191960993814015303, −8.302869917557270503584298453591, −7.38146854312710652449759946027, −6.80763611254033544546906737761, −4.78053949472762705614352328871, −3.60164628060434887919048182840, −1.88652602000414133848387031186, 0.825843150068022389970725750762, 3.28030245159180720631564967496, 4.64922572583733553218386150140, 6.05996994708145097939976443056, 7.26459062538126610816574426785, 8.070880010165438577525148127754, 9.111607141910222670587117415308, 9.907298402465386235823323623719, 11.07206925328739947606845518579, 12.06206006746523401432927087287

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