Properties

Label 2-230-115.9-c1-0-7
Degree $2$
Conductor $230$
Sign $-0.242 + 0.970i$
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 + (−1.78 + 1.54i)3-s + (−0.841 + 0.540i)4-s + (−0.284 − 2.21i)5-s + (−1.98 − 1.27i)6-s + (−3.91 − 1.78i)7-s + (−0.755 − 0.654i)8-s + (0.363 − 2.52i)9-s + (2.04 − 0.897i)10-s + (−1.25 − 0.368i)11-s + (0.663 − 2.26i)12-s + (3.30 − 1.50i)13-s + (0.612 − 4.26i)14-s + (3.92 + 3.51i)15-s + (0.415 − 0.909i)16-s + (−2.01 + 3.12i)17-s + ⋯
L(s)  = 1  + (0.199 + 0.678i)2-s + (−1.02 + 0.890i)3-s + (−0.420 + 0.270i)4-s + (−0.127 − 0.991i)5-s + (−0.809 − 0.520i)6-s + (−1.48 − 0.675i)7-s + (−0.267 − 0.231i)8-s + (0.121 − 0.841i)9-s + (0.647 − 0.283i)10-s + (−0.378 − 0.111i)11-s + (0.191 − 0.652i)12-s + (0.916 − 0.418i)13-s + (0.163 − 1.13i)14-s + (1.01 + 0.906i)15-s + (0.103 − 0.227i)16-s + (−0.487 + 0.758i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.242 + 0.970i$
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: \((2,\ 230,\ (\ :1/2),\ -0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0647029 - 0.0828994i\)
\(L(\frac12)\) \(\approx\) \(0.0647029 - 0.0828994i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.284 + 2.21i)T \)
23 \( 1 + (1.70 + 4.48i)T \)
good3 \( 1 + (1.78 - 1.54i)T + (0.426 - 2.96i)T^{2} \)
7 \( 1 + (3.91 + 1.78i)T + (4.58 + 5.29i)T^{2} \)
11 \( 1 + (1.25 + 0.368i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-3.30 + 1.50i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (2.01 - 3.12i)T + (-7.06 - 15.4i)T^{2} \)
19 \( 1 + (6.15 - 3.95i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (1.95 + 1.25i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-3.09 + 3.56i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (5.75 + 0.827i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (-0.430 - 2.99i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (2.51 - 2.17i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 5.43iT - 47T^{2} \)
53 \( 1 + (-4.88 - 2.23i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (-2.87 - 6.29i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (6.18 - 7.13i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (0.940 + 3.20i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (-12.8 + 3.78i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (7.36 + 11.4i)T + (-30.3 + 66.4i)T^{2} \)
79 \( 1 + (6.84 + 14.9i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-9.90 - 1.42i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (6.13 + 7.08i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-4.82 + 0.693i)T + (93.0 - 27.3i)T^{2} \)
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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

−12.12279804843854713084124972836, −10.66580285888602594746081767821, −10.17558358567636521463835201961, −8.997420045198256619460990039841, −8.001611880204541890195211118071, −6.30593203183303577565043958919, −5.91971376869604972101831566909, −4.49479097937511160678841403142, −3.80623598136747716653243334357, −0.086568937536545741996143920999, 2.30854817213238955392144240168, 3.59683228685767170310876410347, 5.47015888492460930137459083436, 6.51741698839339171998887594327, 6.91481622740968671421590601528, 8.734639137232127803078397567335, 9.865223540009751848238908272740, 10.90103866944135428143512426889, 11.54366112210997360805843401746, 12.40419600347030287423609064746

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