Properties

Label 2-230-115.4-c1-0-0
Degree $2$
Conductor $230$
Sign $-0.631 - 0.775i$
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 + (−0.642 − 2.18i)3-s + (0.654 − 0.755i)4-s + (−1.99 + 1.00i)5-s + (1.49 + 1.72i)6-s + (−2.70 + 0.388i)7-s + (−0.281 + 0.959i)8-s + (−1.85 + 1.19i)9-s + (1.40 − 1.74i)10-s + (−1.73 + 3.80i)11-s + (−2.07 − 0.947i)12-s + (4.03 + 0.579i)13-s + (2.29 − 1.47i)14-s + (3.47 + 3.73i)15-s + (−0.142 − 0.989i)16-s + (−1.45 + 1.25i)17-s + ⋯
L(s)  = 1  + (−0.643 + 0.293i)2-s + (−0.371 − 1.26i)3-s + (0.327 − 0.377i)4-s + (−0.893 + 0.448i)5-s + (0.609 + 0.703i)6-s + (−1.02 + 0.146i)7-s + (−0.0996 + 0.339i)8-s + (−0.618 + 0.397i)9-s + (0.443 − 0.550i)10-s + (−0.524 + 1.14i)11-s + (−0.599 − 0.273i)12-s + (1.11 + 0.160i)13-s + (0.614 − 0.394i)14-s + (0.898 + 0.963i)15-s + (−0.0355 − 0.247i)16-s + (−0.351 + 0.304i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.631 - 0.775i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ -0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0647598 + 0.136199i\)
\(L(\frac12)\) \(\approx\) \(0.0647598 + 0.136199i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.99 - 1.00i)T \)
23 \( 1 + (4.14 - 2.41i)T \)
good3 \( 1 + (0.642 + 2.18i)T + (-2.52 + 1.62i)T^{2} \)
7 \( 1 + (2.70 - 0.388i)T + (6.71 - 1.97i)T^{2} \)
11 \( 1 + (1.73 - 3.80i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-4.03 - 0.579i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (1.45 - 1.25i)T + (2.41 - 16.8i)T^{2} \)
19 \( 1 + (3.27 - 3.77i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (2.14 + 2.46i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (6.90 + 2.02i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-0.263 - 0.410i)T + (-15.3 + 33.6i)T^{2} \)
41 \( 1 + (2.39 + 1.54i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (1.64 + 5.59i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 - 12.6iT - 47T^{2} \)
53 \( 1 + (-0.0981 + 0.0141i)T + (50.8 - 14.9i)T^{2} \)
59 \( 1 + (-1.36 + 9.49i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-4.44 - 1.30i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-10.0 + 4.57i)T + (43.8 - 50.6i)T^{2} \)
71 \( 1 + (-5.81 - 12.7i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (9.10 + 7.88i)T + (10.3 + 72.2i)T^{2} \)
79 \( 1 + (-0.565 + 3.93i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (5.85 + 9.10i)T + (-34.4 + 75.4i)T^{2} \)
89 \( 1 + (4.46 - 1.31i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (4.30 - 6.69i)T + (-40.2 - 88.2i)T^{2} \)
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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

−12.57778318858934064923345479695, −11.65028243457624172139770787874, −10.68040270135031619450667385885, −9.630983684463280382086375778763, −8.271947530544203239245989695927, −7.50794697487975021167549145208, −6.67019360732760608384602462932, −5.95205124409211841050523655244, −3.84057466262550736757598469726, −1.98597154115094583746294733820, 0.15193784623943393554801143385, 3.27715487796161717294210987760, 4.08270531155645459602400552887, 5.51081965751187504416833599142, 6.83769299395059897080492749308, 8.355964223300282624890935654575, 8.953504926860363419678196231768, 10.02632275055950764057636891076, 10.94545354035210597379004153323, 11.32413479522724806770400254253

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