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Properties

Label 2-230-5.4-c3-0-21
Degree $2$
Conductor $230$
Sign $0.973 + 0.230i$
Analytic cond. $13.5704$
Root an. cond. $3.68380$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 3.96i·3-s − 4·4-s + (10.8 + 2.57i)5-s + 7.93·6-s − 5.88i·7-s − 8i·8-s + 11.2·9-s + (−5.15 + 21.7i)10-s − 10.3·11-s + 15.8i·12-s + 1.80i·13-s + 11.7·14-s + (10.2 − 43.1i)15-s + 16·16-s − 74.8i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.763i·3-s − 0.5·4-s + (0.973 + 0.230i)5-s + 0.540·6-s − 0.318i·7-s − 0.353i·8-s + 0.416·9-s + (−0.163 + 0.688i)10-s − 0.283·11-s + 0.381i·12-s + 0.0384i·13-s + 0.224·14-s + (0.176 − 0.743i)15-s + 0.250·16-s − 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ 0.973 + 0.230i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.04113 - 0.238661i\)
\(L(\frac12)\) \(\approx\) \(2.04113 - 0.238661i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (-10.8 - 2.57i)T \)
23 \( 1 + 23iT \)
good3 \( 1 + 3.96iT - 27T^{2} \)
7 \( 1 + 5.88iT - 343T^{2} \)
11 \( 1 + 10.3T + 1.33e3T^{2} \)
13 \( 1 - 1.80iT - 2.19e3T^{2} \)
17 \( 1 + 74.8iT - 4.91e3T^{2} \)
19 \( 1 - 19.0T + 6.85e3T^{2} \)
29 \( 1 - 253.T + 2.43e4T^{2} \)
31 \( 1 - 148.T + 2.97e4T^{2} \)
37 \( 1 + 199. iT - 5.06e4T^{2} \)
41 \( 1 + 26.2T + 6.89e4T^{2} \)
43 \( 1 + 207. iT - 7.95e4T^{2} \)
47 \( 1 + 28.7iT - 1.03e5T^{2} \)
53 \( 1 - 528. iT - 1.48e5T^{2} \)
59 \( 1 + 114.T + 2.05e5T^{2} \)
61 \( 1 - 207.T + 2.26e5T^{2} \)
67 \( 1 - 53.4iT - 3.00e5T^{2} \)
71 \( 1 + 848.T + 3.57e5T^{2} \)
73 \( 1 - 442. iT - 3.89e5T^{2} \)
79 \( 1 + 703.T + 4.93e5T^{2} \)
83 \( 1 - 343. iT - 5.71e5T^{2} \)
89 \( 1 + 642.T + 7.04e5T^{2} \)
97 \( 1 + 665. iT - 9.12e5T^{2} \)
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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

−11.98591984456703637929433749555, −10.50764057726607577756710560268, −9.756700799144149977959452091319, −8.634138402945050808795767703049, −7.40315167606219407027244650454, −6.79181643386783808162348179575, −5.76512569615567344250579609410, −4.55207036929909742313531405787, −2.62174364136136266675400559254, −0.994578325902067307073020804330, 1.42675442765772420288667804252, 2.87897318697442076067675092724, 4.33195797603223307584238468047, 5.27721172199213191754209637124, 6.47294832707163836112192218822, 8.222964464093007755553867414753, 9.164253123020661265269284620610, 10.13080691887100409385472460398, 10.43299091042997260040428602803, 11.76389900017392008101307593363

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