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Properties

Label 2-230-115.114-c4-0-2
Degree $2$
Conductor $230$
Sign $-0.878 - 0.476i$
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  − 2.82i·2-s + 10.7i·3-s − 8.00·4-s + (12.7 − 21.4i)5-s + 30.3·6-s − 32.1·7-s + 22.6i·8-s − 34.1·9-s + (−60.7 − 36.1i)10-s − 66.8i·11-s − 85.8i·12-s + 199. i·13-s + 90.9i·14-s + (230. + 137. i)15-s + 64.0·16-s − 387.·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.19i·3-s − 0.500·4-s + (0.511 − 0.859i)5-s + 0.843·6-s − 0.656·7-s + 0.353i·8-s − 0.422·9-s + (−0.607 − 0.361i)10-s − 0.552i·11-s − 0.596i·12-s + 1.17i·13-s + 0.464i·14-s + (1.02 + 0.609i)15-s + 0.250·16-s − 1.33·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.476i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.878 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.878 - 0.476i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.878 - 0.476i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.2699814371\)
\(L(\frac12)\) \(\approx\) \(0.2699814371\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
5 \( 1 + (-12.7 + 21.4i)T \)
23 \( 1 + (-528. + 20.8i)T \)
good3 \( 1 - 10.7iT - 81T^{2} \)
7 \( 1 + 32.1T + 2.40e3T^{2} \)
11 \( 1 + 66.8iT - 1.46e4T^{2} \)
13 \( 1 - 199. iT - 2.85e4T^{2} \)
17 \( 1 + 387.T + 8.35e4T^{2} \)
19 \( 1 - 79.4iT - 1.30e5T^{2} \)
29 \( 1 + 355.T + 7.07e5T^{2} \)
31 \( 1 + 1.56e3T + 9.23e5T^{2} \)
37 \( 1 + 282.T + 1.87e6T^{2} \)
41 \( 1 + 3.18e3T + 2.82e6T^{2} \)
43 \( 1 + 1.92e3T + 3.41e6T^{2} \)
47 \( 1 + 410. iT - 4.87e6T^{2} \)
53 \( 1 - 397.T + 7.89e6T^{2} \)
59 \( 1 - 430.T + 1.21e7T^{2} \)
61 \( 1 - 3.71e3iT - 1.38e7T^{2} \)
67 \( 1 + 6.25e3T + 2.01e7T^{2} \)
71 \( 1 - 4.73e3T + 2.54e7T^{2} \)
73 \( 1 - 7.51e3iT - 2.83e7T^{2} \)
79 \( 1 + 3.31e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.16e4T + 4.74e7T^{2} \)
89 \( 1 + 9.23e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.18e4T + 8.85e7T^{2} \)
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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

−11.75743666940915635864135868757, −10.88141956317189458629744068984, −9.961206576644749571947981163574, −9.132610489062135553379135974168, −8.763350859154856159973240262042, −6.74010215215516695480245014820, −5.34146894131016688341520456572, −4.44188396746370500232003317887, −3.45480148166561984446027712337, −1.78556561751314371161337672607, 0.085635861154761068113670234932, 1.90995757632782313070926870391, 3.29149499399447660973510270739, 5.20281219964529869350530924862, 6.44944777230393896790567108856, 6.90944716331392284940146942523, 7.75887874815784524601128390223, 9.042017395561883403256786206951, 10.08892614610974406190320616759, 11.10664237246459846365134838267

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