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Properties

Label 2-230-5.2-c4-0-3
Degree $2$
Conductor $230$
Sign $0.628 + 0.777i$
Analytic cond. $23.7750$
Root an. cond. $4.87597$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 2i)2-s + (−11.0 + 11.0i)3-s + 8i·4-s + (3.15 + 24.8i)5-s − 44.1·6-s + (−2.46 − 2.46i)7-s + (−16 + 16i)8-s − 163. i·9-s + (−43.2 + 55.9i)10-s − 221.·11-s + (−88.3 − 88.3i)12-s + (98.6 − 98.6i)13-s − 9.85i·14-s + (−308. − 239. i)15-s − 64·16-s + (17.4 + 17.4i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−1.22 + 1.22i)3-s + 0.5i·4-s + (0.126 + 0.992i)5-s − 1.22·6-s + (−0.0502 − 0.0502i)7-s + (−0.250 + 0.250i)8-s − 2.01i·9-s + (−0.432 + 0.559i)10-s − 1.83·11-s + (−0.613 − 0.613i)12-s + (0.583 − 0.583i)13-s − 0.0502i·14-s + (−1.37 − 1.06i)15-s − 0.250·16-s + (0.0604 + 0.0604i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.628 + 0.777i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.628 + 0.777i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1952845711\)
\(L(\frac12)\) \(\approx\) \(0.1952845711\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 2i)T \)
5 \( 1 + (-3.15 - 24.8i)T \)
23 \( 1 + (-77.9 + 77.9i)T \)
good3 \( 1 + (11.0 - 11.0i)T - 81iT^{2} \)
7 \( 1 + (2.46 + 2.46i)T + 2.40e3iT^{2} \)
11 \( 1 + 221.T + 1.46e4T^{2} \)
13 \( 1 + (-98.6 + 98.6i)T - 2.85e4iT^{2} \)
17 \( 1 + (-17.4 - 17.4i)T + 8.35e4iT^{2} \)
19 \( 1 - 413. iT - 1.30e5T^{2} \)
29 \( 1 - 453. iT - 7.07e5T^{2} \)
31 \( 1 - 1.07e3T + 9.23e5T^{2} \)
37 \( 1 + (1.65e3 + 1.65e3i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.55e3T + 2.82e6T^{2} \)
43 \( 1 + (631. - 631. i)T - 3.41e6iT^{2} \)
47 \( 1 + (-644. - 644. i)T + 4.87e6iT^{2} \)
53 \( 1 + (-793. + 793. i)T - 7.89e6iT^{2} \)
59 \( 1 - 2.17e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.37e3T + 1.38e7T^{2} \)
67 \( 1 + (-1.21e3 - 1.21e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 8.01e3T + 2.54e7T^{2} \)
73 \( 1 + (5.46e3 - 5.46e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 5.75e3iT - 3.89e7T^{2} \)
83 \( 1 + (7.12e3 - 7.12e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 302. iT - 6.27e7T^{2} \)
97 \( 1 + (5.42e3 + 5.42e3i)T + 8.85e7iT^{2} \)
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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

−12.27621473937374393390428089537, −11.17067819618181862798172971999, −10.48154181314529280458335078299, −9.998074231110290249139851528655, −8.303677945811586719587904523282, −7.08302941681698814221699238990, −5.82210533244858272700678127091, −5.43198524408729899343496801726, −4.09316065319301467526706224655, −2.97053895574226688944467584701, 0.07122510018623997684111609979, 1.17045278361637383266897619800, 2.46440214570370731461423946679, 4.69338122810104957620189227772, 5.39207648312418491665008276827, 6.33210062077573996681778184411, 7.51521877457061326695675210412, 8.615009636539546474401018420400, 10.11503743985858344291678621774, 11.08896697840691663941537229764

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