Properties

Label 2-230-115.9-c1-0-8
Degree $2$
Conductor $230$
Sign $0.000240 + 0.999i$
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  + (−0.281 − 0.959i)2-s + (2.36 − 2.05i)3-s + (−0.841 + 0.540i)4-s + (1.30 + 1.81i)5-s + (−2.63 − 1.69i)6-s + (−1.55 − 0.710i)7-s + (0.755 + 0.654i)8-s + (0.972 − 6.76i)9-s + (1.37 − 1.76i)10-s + (−0.297 − 0.0874i)11-s + (−0.883 + 3.00i)12-s + (2.61 − 1.19i)13-s + (−0.243 + 1.69i)14-s + (6.81 + 1.63i)15-s + (0.415 − 0.909i)16-s + (−2.98 + 4.65i)17-s + ⋯
L(s)  = 1  + (−0.199 − 0.678i)2-s + (1.36 − 1.18i)3-s + (−0.420 + 0.270i)4-s + (0.581 + 0.813i)5-s + (−1.07 − 0.692i)6-s + (−0.588 − 0.268i)7-s + (0.267 + 0.231i)8-s + (0.324 − 2.25i)9-s + (0.435 − 0.556i)10-s + (−0.0897 − 0.0263i)11-s + (−0.254 + 0.868i)12-s + (0.723 − 0.330i)13-s + (−0.0650 + 0.452i)14-s + (1.76 + 0.422i)15-s + (0.103 − 0.227i)16-s + (−0.725 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.000240 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.000240 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.000240 + 0.999i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.000240 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16025 - 1.15997i\)
\(L(\frac12)\) \(\approx\) \(1.16025 - 1.15997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.281 + 0.959i)T \)
5 \( 1 + (-1.30 - 1.81i)T \)
23 \( 1 + (0.555 + 4.76i)T \)
good3 \( 1 + (-2.36 + 2.05i)T + (0.426 - 2.96i)T^{2} \)
7 \( 1 + (1.55 + 0.710i)T + (4.58 + 5.29i)T^{2} \)
11 \( 1 + (0.297 + 0.0874i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-2.61 + 1.19i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (2.98 - 4.65i)T + (-7.06 - 15.4i)T^{2} \)
19 \( 1 + (3.72 - 2.39i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-7.45 - 4.79i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (5.47 - 6.31i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-5.64 - 0.811i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.124 + 0.869i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (3.72 - 3.22i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + (-3.19 - 1.46i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (-2.22 - 4.86i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (3.37 - 3.89i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-0.478 - 1.62i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (-7.87 + 2.31i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (1.64 + 2.55i)T + (-30.3 + 66.4i)T^{2} \)
79 \( 1 + (3.42 + 7.49i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-0.701 - 0.100i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (-6.06 - 6.99i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (17.0 - 2.45i)T + (93.0 - 27.3i)T^{2} \)
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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

−12.33911835936432445583322641966, −10.78004846305802993180033044984, −10.10671530377288862843390058280, −8.814604778314159021121760690432, −8.295045210607606118454443330555, −6.94833128783279744557191598417, −6.28298244077225155177605506627, −3.75831747385565278345503021561, −2.81216919895747624647365203105, −1.68693866590954465790925542728, 2.46852895165724080603589459803, 4.04863056863715278429502629393, 4.96519815635428514156285879245, 6.29432848680309362638457600284, 7.85718407383540512791224388918, 8.786428374752849806767073518740, 9.353044891002732948108644194821, 9.894187199054946950771342893513, 11.20630645229523976532670509247, 12.99348745814384963492925348653

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