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Properties

Label 2-230-5.2-c4-0-1
Degree $2$
Conductor $230$
Sign $-0.215 + 0.976i$
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 + (−5.15 + 5.15i)3-s + 8i·4-s + (−17.4 + 17.9i)5-s + 20.6·6-s + (−7.30 − 7.30i)7-s + (16 − 16i)8-s + 27.9i·9-s + (70.7 − 1.02i)10-s − 194.·11-s + (−41.2 − 41.2i)12-s + (−142. + 142. i)13-s + 29.2i·14-s + (−2.63 − 182. i)15-s − 64·16-s + (281. + 281. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.572 + 0.572i)3-s + 0.5i·4-s + (−0.696 + 0.717i)5-s + 0.572·6-s + (−0.149 − 0.149i)7-s + (0.250 − 0.250i)8-s + 0.344i·9-s + (0.707 − 0.0102i)10-s − 1.60·11-s + (−0.286 − 0.286i)12-s + (−0.841 + 0.841i)13-s + 0.149i·14-s + (−0.0117 − 0.809i)15-s − 0.250·16-s + (0.974 + 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.215 + 0.976i$
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: \((2,\ 230,\ (\ :2),\ -0.215 + 0.976i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.02962500124\)
\(L(\frac12)\) \(\approx\) \(0.02962500124\)
\(L(3)\) 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 \( 1 + (17.4 - 17.9i)T \)
23 \( 1 + (77.9 - 77.9i)T \)
good3 \( 1 + (5.15 - 5.15i)T - 81iT^{2} \)
7 \( 1 + (7.30 + 7.30i)T + 2.40e3iT^{2} \)
11 \( 1 + 194.T + 1.46e4T^{2} \)
13 \( 1 + (142. - 142. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-281. - 281. i)T + 8.35e4iT^{2} \)
19 \( 1 - 479. iT - 1.30e5T^{2} \)
29 \( 1 + 968. iT - 7.07e5T^{2} \)
31 \( 1 - 293.T + 9.23e5T^{2} \)
37 \( 1 + (256. + 256. i)T + 1.87e6iT^{2} \)
41 \( 1 - 2.02e3T + 2.82e6T^{2} \)
43 \( 1 + (2.24e3 - 2.24e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-478. - 478. i)T + 4.87e6iT^{2} \)
53 \( 1 + (1.37e3 - 1.37e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 5.40e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.14e3T + 1.38e7T^{2} \)
67 \( 1 + (3.54e3 + 3.54e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 1.70e3T + 2.54e7T^{2} \)
73 \( 1 + (76.1 - 76.1i)T - 2.83e7iT^{2} \)
79 \( 1 - 89.8iT - 3.89e7T^{2} \)
83 \( 1 + (-4.48e3 + 4.48e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.45e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.54e3 + 1.54e3i)T + 8.85e7iT^{2} \)
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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

−11.93652311361917807851362850924, −11.10705644590249910975764246169, −10.23892816262778303727631460479, −9.914592918539349455048459536840, −8.013021814269027188921593879859, −7.68760699437085184511948587873, −6.09711091189663723408800746279, −4.76254768080969617228263661577, −3.59872900779479775344588534337, −2.23055881839537517848477698897, 0.01841941982553304146844166725, 0.792260187651210304738093363751, 2.89368209392102968926746241456, 4.96741094629270292043087887052, 5.54607339634714727610535245027, 7.11429200289806214183414399331, 7.63816385545832246208243534224, 8.719940934962930341355807609815, 9.784663325084023044593615289881, 10.85599813310823085946294320554

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