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Properties

Label 2-230-115.114-c2-0-21
Degree $2$
Conductor $230$
Sign $-0.933 - 0.359i$
Analytic cond. $6.26704$
Root an. cond. $2.50340$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 5.45i·3-s − 2.00·4-s + (3.90 − 3.12i)5-s − 7.71·6-s − 0.770·7-s + 2.82i·8-s − 20.7·9-s + (−4.41 − 5.52i)10-s − 8.45i·11-s + 10.9i·12-s − 4.15i·13-s + 1.09i·14-s + (−17.0 − 21.2i)15-s + 4.00·16-s + 32.2·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.81i·3-s − 0.500·4-s + (0.781 − 0.624i)5-s − 1.28·6-s − 0.110·7-s + 0.353i·8-s − 2.30·9-s + (−0.441 − 0.552i)10-s − 0.768i·11-s + 0.908i·12-s − 0.319i·13-s + 0.0778i·14-s + (−1.13 − 1.41i)15-s + 0.250·16-s + 1.89·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.933 - 0.359i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ -0.933 - 0.359i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.274681 + 1.47725i\)
\(L(\frac12)\) \(\approx\) \(0.274681 + 1.47725i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
5 \( 1 + (-3.90 + 3.12i)T \)
23 \( 1 + (19.8 - 11.6i)T \)
good3 \( 1 + 5.45iT - 9T^{2} \)
7 \( 1 + 0.770T + 49T^{2} \)
11 \( 1 + 8.45iT - 121T^{2} \)
13 \( 1 + 4.15iT - 169T^{2} \)
17 \( 1 - 32.2T + 289T^{2} \)
19 \( 1 - 23.8iT - 361T^{2} \)
29 \( 1 - 21.7T + 841T^{2} \)
31 \( 1 + 31.8T + 961T^{2} \)
37 \( 1 - 6.03T + 1.36e3T^{2} \)
41 \( 1 - 16.9T + 1.68e3T^{2} \)
43 \( 1 - 61.7T + 1.84e3T^{2} \)
47 \( 1 + 45.0iT - 2.20e3T^{2} \)
53 \( 1 - 45.8T + 2.80e3T^{2} \)
59 \( 1 + 93.6T + 3.48e3T^{2} \)
61 \( 1 + 37.1iT - 3.72e3T^{2} \)
67 \( 1 + 44.2T + 4.48e3T^{2} \)
71 \( 1 - 77.7T + 5.04e3T^{2} \)
73 \( 1 + 101. iT - 5.32e3T^{2} \)
79 \( 1 + 14.4iT - 6.24e3T^{2} \)
83 \( 1 + 44.9T + 6.88e3T^{2} \)
89 \( 1 - 74.0iT - 7.92e3T^{2} \)
97 \( 1 + 42.2T + 9.40e3T^{2} \)
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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

−11.95111260612342889476603124423, −10.58599603017681046598616297441, −9.478641344529125740911610386452, −8.295059300523409202550488981017, −7.67923498083829298578602455424, −6.05710479175324377898276029407, −5.56333723627784802251184693075, −3.29057092045563200576673332407, −1.86483515213911196281564220682, −0.862840464660395701746573805924, 2.91596557478082577541300186562, 4.21874429098402116918623098907, 5.25200350045414303205399480430, 6.14250547390599568553352736410, 7.51580178207362412496972443781, 8.950030996810338598660850055096, 9.697478846112407085997059007441, 10.22557292099055985569424694056, 11.18326393347687996709026058374, 12.53422182271545452565325003063

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