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Properties

Label 2-230-23.22-c4-0-17
Degree $2$
Conductor $230$
Sign $0.783 + 0.621i$
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  − 2.82·2-s + 0.828·3-s + 8.00·4-s − 11.1i·5-s − 2.34·6-s + 16.9i·7-s − 22.6·8-s − 80.3·9-s + 31.6i·10-s + 34.1i·11-s + 6.62·12-s + 274.·13-s − 47.8i·14-s − 9.26i·15-s + 64.0·16-s − 6.91i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.0920·3-s + 0.500·4-s − 0.447i·5-s − 0.0651·6-s + 0.345i·7-s − 0.353·8-s − 0.991·9-s + 0.316i·10-s + 0.281i·11-s + 0.0460·12-s + 1.62·13-s − 0.244i·14-s − 0.0411i·15-s + 0.250·16-s − 0.0239i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.783 + 0.621i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.783 + 0.621i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.269494641\)
\(L(\frac12)\) \(\approx\) \(1.269494641\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82T \)
5 \( 1 + 11.1iT \)
23 \( 1 + (328. - 414. i)T \)
good3 \( 1 - 0.828T + 81T^{2} \)
7 \( 1 - 16.9iT - 2.40e3T^{2} \)
11 \( 1 - 34.1iT - 1.46e4T^{2} \)
13 \( 1 - 274.T + 2.85e4T^{2} \)
17 \( 1 + 6.91iT - 8.35e4T^{2} \)
19 \( 1 + 302. iT - 1.30e5T^{2} \)
29 \( 1 - 266.T + 7.07e5T^{2} \)
31 \( 1 - 583.T + 9.23e5T^{2} \)
37 \( 1 + 1.13e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.76e3T + 2.82e6T^{2} \)
43 \( 1 + 3.30e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.72e3T + 4.87e6T^{2} \)
53 \( 1 + 2.67e3iT - 7.89e6T^{2} \)
59 \( 1 - 1.27e3T + 1.21e7T^{2} \)
61 \( 1 - 4.18e3iT - 1.38e7T^{2} \)
67 \( 1 + 6.23e3iT - 2.01e7T^{2} \)
71 \( 1 - 7.81e3T + 2.54e7T^{2} \)
73 \( 1 - 5.45e3T + 2.83e7T^{2} \)
79 \( 1 + 1.15e4iT - 3.89e7T^{2} \)
83 \( 1 + 2.46e3iT - 4.74e7T^{2} \)
89 \( 1 + 242. iT - 6.27e7T^{2} \)
97 \( 1 + 1.36e4iT - 8.85e7T^{2} \)
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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

−11.37182336943148993410289538965, −10.49851900918114098386658751091, −9.148945903060906782471849481654, −8.716032653673154001396489647341, −7.71125379269689312701582154132, −6.33114103992499431729856305594, −5.42147852497560401215872739952, −3.71723396753014084407717287525, −2.24638178582481989684582015034, −0.68791299821927291480326086302, 1.00245406795340152871594093284, 2.68873606245815556581676079668, 3.90042089661208889567016720271, 5.84040160688212556197996955966, 6.53327164985499802062003344587, 8.011735188764447814095823464503, 8.513578401173476909701909189784, 9.721461713610847463910853807689, 10.77931456859867821577075710304, 11.27826831373295832094812344348

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