↑

Properties

Label 2-230-115.114-c4-0-11
Degree $2$
Conductor $230$
Sign $0.779 - 0.625i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s − 15.5i·3-s − 8.00·4-s + (−22.6 + 10.6i)5-s + 43.9·6-s + 25.0·7-s − 22.6i·8-s − 160.·9-s + (−30.1 − 63.9i)10-s + 156. i·11-s + 124. i·12-s − 61.5i·13-s + 70.8i·14-s + (165. + 351. i)15-s + 64.0·16-s − 38.3·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.72i·3-s − 0.500·4-s + (−0.904 + 0.426i)5-s + 1.22·6-s + 0.511·7-s − 0.353i·8-s − 1.98·9-s + (−0.301 − 0.639i)10-s + 1.29i·11-s + 0.863i·12-s − 0.364i·13-s + 0.361i·14-s + (0.736 + 1.56i)15-s + 0.250·16-s − 0.132·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.625i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.779 - 0.625i$
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: \((2,\ 230,\ (\ :2),\ 0.779 - 0.625i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.246166781\)
\(L(\frac12)\) \(\approx\) \(1.246166781\)
\(L(3)\) 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 \( 1 + (22.6 - 10.6i)T \)
23 \( 1 + (-123. - 514. i)T \)
good3 \( 1 + 15.5iT - 81T^{2} \)
7 \( 1 - 25.0T + 2.40e3T^{2} \)
11 \( 1 - 156. iT - 1.46e4T^{2} \)
13 \( 1 + 61.5iT - 2.85e4T^{2} \)
17 \( 1 + 38.3T + 8.35e4T^{2} \)
19 \( 1 + 19.8iT - 1.30e5T^{2} \)
29 \( 1 + 1.10e3T + 7.07e5T^{2} \)
31 \( 1 - 1.30e3T + 9.23e5T^{2} \)
37 \( 1 - 2.52e3T + 1.87e6T^{2} \)
41 \( 1 - 1.79e3T + 2.82e6T^{2} \)
43 \( 1 - 3.19e3T + 3.41e6T^{2} \)
47 \( 1 - 1.73e3iT - 4.87e6T^{2} \)
53 \( 1 - 98.3T + 7.89e6T^{2} \)
59 \( 1 - 2.86e3T + 1.21e7T^{2} \)
61 \( 1 - 2.29e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.97e3T + 2.01e7T^{2} \)
71 \( 1 + 3.01e3T + 2.54e7T^{2} \)
73 \( 1 - 2.80e3iT - 2.83e7T^{2} \)
79 \( 1 - 6.13e3iT - 3.89e7T^{2} \)
83 \( 1 - 5.32e3T + 4.74e7T^{2} \)
89 \( 1 - 5.44e3iT - 6.27e7T^{2} \)
97 \( 1 - 9.24e3T + 8.85e7T^{2} \)
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

−11.87621936916555302529870840212, −11.02038152005158312256407999402, −9.397614527750393089721764723931, −7.997106401012400799601402731465, −7.63574952715710980492442164820, −6.91131746177924300295011016198, −5.82415399531898985915399273188, −4.34831641135812290945500166807, −2.55037875791219138569956863159, −1.01735895261131806680528067802, 0.54483238934493021797232259616, 2.92344999770865230932356592309, 4.04168604077333063412142831516, 4.62680207199950052197359401264, 5.81493282036375653723396101588, 7.983169577236135858310186044793, 8.789814314012596470372494339826, 9.509011381101891069838703258094, 10.79420655222448701623346564167, 11.14173177962803064039065178859

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