Properties

Label 2-230-115.29-c1-0-4
Degree $2$
Conductor $230$
Sign $0.407 - 0.913i$
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 + (−1.78 + 1.34i)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 + (−2.18 + 0.487i)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 + (−0.101 − 0.267i)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 + (−0.797 + 0.603i)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 + (−0.690 + 0.154i)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 + (−0.0261 − 0.0689i)15-s + (−0.0355 + 0.247i)16-s + (−1.40 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.407 - 0.913i$
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: \((2,\ 230,\ (\ :1/2),\ 0.407 - 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36834 + 0.888248i\)
\(L(\frac12)\) \(\approx\) \(1.36834 + 0.888248i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.78 - 1.34i)T \)
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

−12.52264800209957691999341673910, −11.37456687995156333271235549001, −10.81551797187430408622482625592, −9.480450834273447992019772949056, −8.192624414450432862030936815321, −7.07808349487622073923528940306, −6.58143772034018478503301604613, −4.67587474936546925386261968613, −4.14046586755876462404530840112, −2.40510781405899034729449825144, 1.36411689778273897670276368372, 3.71357402128797328941023911053, 4.17241738502015352792848156834, 5.85083176861139730790755506786, 6.71214602574709265366387273175, 8.241838374225183994449037747223, 8.922723458670411916463504011281, 10.37915441723146379358028168857, 11.37890752604572110702005176749, 11.93112007033593357200863227273

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