↑

Properties

Label 2-230-5.4-c5-0-46
Degree $2$
Conductor $230$
Sign $-0.759 + 0.649i$
Analytic cond. $36.8882$
Root an. cond. $6.07357$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 11.0i·3-s − 16·4-s + (−42.4 + 36.3i)5-s + 44.3·6-s − 240. i·7-s − 64i·8-s + 120.·9-s + (−145. − 169. i)10-s + 467.·11-s + 177. i·12-s + 75.0i·13-s + 963.·14-s + (402. + 470. i)15-s + 256·16-s + 1.16e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.711i·3-s − 0.5·4-s + (−0.759 + 0.649i)5-s + 0.502·6-s − 1.85i·7-s − 0.353i·8-s + 0.494·9-s + (−0.459 − 0.537i)10-s + 1.16·11-s + 0.355i·12-s + 0.123i·13-s + 1.31·14-s + (0.462 + 0.540i)15-s + 0.250·16-s + 0.980i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.649i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.759 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.759 + 0.649i$
Analytic conductor: \(36.8882\)
Root analytic conductor: \(6.07357\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :5/2),\ -0.759 + 0.649i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6483875903\)
\(L(\frac12)\) \(\approx\) \(0.6483875903\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 + (42.4 - 36.3i)T \)
23 \( 1 - 529iT \)
good3 \( 1 + 11.0iT - 243T^{2} \)
7 \( 1 + 240. iT - 1.68e4T^{2} \)
11 \( 1 - 467.T + 1.61e5T^{2} \)
13 \( 1 - 75.0iT - 3.71e5T^{2} \)
17 \( 1 - 1.16e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.64e3T + 2.47e6T^{2} \)
29 \( 1 - 4.64e3T + 2.05e7T^{2} \)
31 \( 1 + 4.03e3T + 2.86e7T^{2} \)
37 \( 1 + 7.19e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.68e4T + 1.15e8T^{2} \)
43 \( 1 + 2.14e4iT - 1.47e8T^{2} \)
47 \( 1 - 7.46e3iT - 2.29e8T^{2} \)
53 \( 1 + 2.67e3iT - 4.18e8T^{2} \)
59 \( 1 - 1.68e4T + 7.14e8T^{2} \)
61 \( 1 + 5.46e4T + 8.44e8T^{2} \)
67 \( 1 - 4.14e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.19e4T + 1.80e9T^{2} \)
73 \( 1 - 1.01e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.88e4T + 3.07e9T^{2} \)
83 \( 1 - 2.40e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.90e4T + 5.58e9T^{2} \)
97 \( 1 + 1.74e5iT - 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.78045909485283567964469406925, −10.20736511868911157295860474724, −8.613408336289276844625123099468, −7.64400643006395149172263784514, −6.87631101648740375325682310331, −6.47089814120766112627736680068, −4.24051113671857053344454470229, −3.85204692396573084825549755858, −1.49330827272096297987910942691, −0.19880321200947849975181186454, 1.59620651067887315376577729516, 3.07103450822149430261587834967, 4.33701442195461046019889842851, 5.06259697136986259754199939664, 6.49814334643615339592922618824, 8.275694002189865221861418221683, 8.991140263873143274583741743480, 9.587674224085379069885669026934, 10.88498059735110169016034723685, 11.91416377598859792138560844014

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