Properties

Label 2-230-5.3-c4-0-25
Degree $2$
Conductor $230$
Sign $-0.457 + 0.889i$
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 + (−8.96 − 8.96i)3-s − 8i·4-s + (1.96 + 24.9i)5-s − 35.8·6-s + (53.1 − 53.1i)7-s + (−16 − 16i)8-s + 79.8i·9-s + (53.7 + 45.9i)10-s + 151.·11-s + (−71.7 + 71.7i)12-s + (162. + 162. i)13-s − 212. i·14-s + (205. − 241. i)15-s − 64·16-s + (130. − 130. i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.996 − 0.996i)3-s − 0.5i·4-s + (0.0787 + 0.996i)5-s − 0.996·6-s + (1.08 − 1.08i)7-s + (−0.250 − 0.250i)8-s + 0.986i·9-s + (0.537 + 0.459i)10-s + 1.24·11-s + (−0.498 + 0.498i)12-s + (0.960 + 0.960i)13-s − 1.08i·14-s + (0.915 − 1.07i)15-s − 0.250·16-s + (0.450 − 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.457 + 0.889i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.457 + 0.889i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.213226842\)
\(L(\frac12)\) \(\approx\) \(2.213226842\)
\(L(3)\) 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 \( 1 + (-1.96 - 24.9i)T \)
23 \( 1 + (-77.9 - 77.9i)T \)
good3 \( 1 + (8.96 + 8.96i)T + 81iT^{2} \)
7 \( 1 + (-53.1 + 53.1i)T - 2.40e3iT^{2} \)
11 \( 1 - 151.T + 1.46e4T^{2} \)
13 \( 1 + (-162. - 162. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-130. + 130. i)T - 8.35e4iT^{2} \)
19 \( 1 + 340. iT - 1.30e5T^{2} \)
29 \( 1 + 797. iT - 7.07e5T^{2} \)
31 \( 1 - 1.24e3T + 9.23e5T^{2} \)
37 \( 1 + (42.4 - 42.4i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.54e3T + 2.82e6T^{2} \)
43 \( 1 + (100. + 100. i)T + 3.41e6iT^{2} \)
47 \( 1 + (2.15e3 - 2.15e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (2.12e3 + 2.12e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 6.38e3iT - 1.21e7T^{2} \)
61 \( 1 + 826.T + 1.38e7T^{2} \)
67 \( 1 + (-3.95e3 + 3.95e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 5.69e3T + 2.54e7T^{2} \)
73 \( 1 + (-5.55e3 - 5.55e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 2.29e3iT - 3.89e7T^{2} \)
83 \( 1 + (5.24e3 + 5.24e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.35e3iT - 6.27e7T^{2} \)
97 \( 1 + (-4.85e3 + 4.85e3i)T - 8.85e7iT^{2} \)
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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

−11.57519590804114651815353309396, −10.84096246828144186346494304011, −9.590401118105800404862167784947, −7.901550641555010588895981195190, −6.68449847554845750988505509969, −6.46349461412844836436104575482, −4.85636703683496694499524821957, −3.68416203513899515906914788850, −1.79462983183435606662052734184, −0.863613895279208122097943165258, 1.34753573750148970789193515099, 3.74094116660397899920812933944, 4.79614247810040024613862518899, 5.51648407158568004649656012598, 6.18923871486477449935728222750, 8.177446359347695909886894669137, 8.726793252955034338279456830418, 9.956503751372785778219084618333, 11.13571122727875228753696225406, 11.94392390858964953226569557096

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