Properties

Label 2-230-115.109-c2-0-4
Degree $2$
Conductor $230$
Sign $-0.800 - 0.598i$
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  + (1.28 + 0.587i)2-s + (−0.540 + 1.84i)3-s + (1.30 + 1.51i)4-s + (−3.54 + 3.52i)5-s + (−1.77 + 2.05i)6-s + (0.0656 − 0.456i)7-s + (0.796 + 2.71i)8-s + (4.47 + 2.87i)9-s + (−6.63 + 2.44i)10-s + (0.696 − 0.318i)11-s + (−3.49 + 1.59i)12-s + (−20.8 + 2.99i)13-s + (0.352 − 0.549i)14-s + (−4.56 − 8.43i)15-s + (−0.569 + 3.95i)16-s + (2.37 − 2.74i)17-s + ⋯
L(s)  = 1  + (0.643 + 0.293i)2-s + (−0.180 + 0.613i)3-s + (0.327 + 0.377i)4-s + (−0.709 + 0.704i)5-s + (−0.296 + 0.341i)6-s + (0.00938 − 0.0652i)7-s + (0.0996 + 0.339i)8-s + (0.496 + 0.319i)9-s + (−0.663 + 0.244i)10-s + (0.0633 − 0.0289i)11-s + (−0.290 + 0.132i)12-s + (−1.60 + 0.230i)13-s + (0.0252 − 0.0392i)14-s + (−0.304 − 0.562i)15-s + (−0.0355 + 0.247i)16-s + (0.139 − 0.161i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.800 - 0.598i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ -0.800 - 0.598i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.499986 + 1.50316i\)
\(L(\frac12)\) \(\approx\) \(0.499986 + 1.50316i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.587i)T \)
5 \( 1 + (3.54 - 3.52i)T \)
23 \( 1 + (-9.91 - 20.7i)T \)
good3 \( 1 + (0.540 - 1.84i)T + (-7.57 - 4.86i)T^{2} \)
7 \( 1 + (-0.0656 + 0.456i)T + (-47.0 - 13.8i)T^{2} \)
11 \( 1 + (-0.696 + 0.318i)T + (79.2 - 91.4i)T^{2} \)
13 \( 1 + (20.8 - 2.99i)T + (162. - 47.6i)T^{2} \)
17 \( 1 + (-2.37 + 2.74i)T + (-41.1 - 286. i)T^{2} \)
19 \( 1 + (22.1 - 19.1i)T + (51.3 - 357. i)T^{2} \)
29 \( 1 + (-26.9 + 31.1i)T + (-119. - 832. i)T^{2} \)
31 \( 1 + (-35.8 + 10.5i)T + (808. - 519. i)T^{2} \)
37 \( 1 + (-57.6 - 37.0i)T + (568. + 1.24e3i)T^{2} \)
41 \( 1 + (30.9 - 19.9i)T + (698. - 1.52e3i)T^{2} \)
43 \( 1 + (-31.1 - 9.14i)T + (1.55e3 + 9.99e2i)T^{2} \)
47 \( 1 + 60.7iT - 2.20e3T^{2} \)
53 \( 1 + (11.2 - 78.3i)T + (-2.69e3 - 791. i)T^{2} \)
59 \( 1 + (8.45 + 58.7i)T + (-3.33e3 + 980. i)T^{2} \)
61 \( 1 + (-30.8 - 105. i)T + (-3.13e3 + 2.01e3i)T^{2} \)
67 \( 1 + (-45.4 + 99.4i)T + (-2.93e3 - 3.39e3i)T^{2} \)
71 \( 1 + (33.8 - 74.0i)T + (-3.30e3 - 3.80e3i)T^{2} \)
73 \( 1 + (-72.5 + 62.8i)T + (758. - 5.27e3i)T^{2} \)
79 \( 1 + (21.5 - 3.09i)T + (5.98e3 - 1.75e3i)T^{2} \)
83 \( 1 + (75.4 + 48.4i)T + (2.86e3 + 6.26e3i)T^{2} \)
89 \( 1 + (-5.79 + 19.7i)T + (-6.66e3 - 4.28e3i)T^{2} \)
97 \( 1 + (-58.1 + 37.3i)T + (3.90e3 - 8.55e3i)T^{2} \)
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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

−12.20694347426497524103806068332, −11.57140005652070139030178262248, −10.43302053765855377756657981359, −9.775491158206661311487765054424, −8.071023409593910592755825508888, −7.31944886776235187557611095886, −6.22907090372843876527214714989, −4.76561784409211835611226306620, −4.07051611844256416131217120450, −2.60308369791842350244772651248, 0.71322133917808364438960200307, 2.51798369148252717638704478958, 4.24218012523968573651498984479, 5.02447439044009472284994393843, 6.56072397570142581696792003836, 7.36253580979870374529377414957, 8.561293967067496006459600972177, 9.755498467219426691474698001513, 10.89960107294693540270101128783, 11.98766529643672956100777363729

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