↑

Properties

Label 2-230-5.3-c2-0-12
Degree $2$
Conductor $230$
Sign $0.997 + 0.0725i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (2.98 + 2.98i)3-s − 2i·4-s + (4.43 − 2.31i)5-s + 5.96·6-s + (−2.75 + 2.75i)7-s + (−2 − 2i)8-s + 8.80i·9-s + (2.11 − 6.74i)10-s + 18.9·11-s + (5.96 − 5.96i)12-s + (−10.4 − 10.4i)13-s + 5.50i·14-s + (20.1 + 6.32i)15-s − 4·16-s + (−14.4 + 14.4i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.994 + 0.994i)3-s − 0.5i·4-s + (0.886 − 0.462i)5-s + 0.994·6-s + (−0.393 + 0.393i)7-s + (−0.250 − 0.250i)8-s + 0.977i·9-s + (0.211 − 0.674i)10-s + 1.72·11-s + (0.497 − 0.497i)12-s + (−0.800 − 0.800i)13-s + 0.393i·14-s + (1.34 + 0.421i)15-s − 0.250·16-s + (−0.851 + 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.997 + 0.0725i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.997 + 0.0725i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.01474 - 0.109500i\)
\(L(\frac12)\) \(\approx\) \(3.01474 - 0.109500i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + (-4.43 + 2.31i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good3 \( 1 + (-2.98 - 2.98i)T + 9iT^{2} \)
7 \( 1 + (2.75 - 2.75i)T - 49iT^{2} \)
11 \( 1 - 18.9T + 121T^{2} \)
13 \( 1 + (10.4 + 10.4i)T + 169iT^{2} \)
17 \( 1 + (14.4 - 14.4i)T - 289iT^{2} \)
19 \( 1 - 6.41iT - 361T^{2} \)
29 \( 1 - 47.3iT - 841T^{2} \)
31 \( 1 + 30.5T + 961T^{2} \)
37 \( 1 + (-33.0 + 33.0i)T - 1.36e3iT^{2} \)
41 \( 1 + 29.3T + 1.68e3T^{2} \)
43 \( 1 + (48.4 + 48.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-31.5 + 31.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (67.5 + 67.5i)T + 2.80e3iT^{2} \)
59 \( 1 - 14.7iT - 3.48e3T^{2} \)
61 \( 1 + 69.1T + 3.72e3T^{2} \)
67 \( 1 + (64.7 - 64.7i)T - 4.48e3iT^{2} \)
71 \( 1 - 48.8T + 5.04e3T^{2} \)
73 \( 1 + (-41.0 - 41.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 25.2iT - 6.24e3T^{2} \)
83 \( 1 + (-3.17 - 3.17i)T + 6.88e3iT^{2} \)
89 \( 1 - 10.7iT - 7.92e3T^{2} \)
97 \( 1 + (-9.11 + 9.11i)T - 9.40e3iT^{2} \)
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

−12.22346418319309447362881210076, −10.77968944631512633026426408737, −9.891127442315603674610934185463, −9.199385606147248576021064041382, −8.612057244606934314945750807886, −6.65460086359677355168254740460, −5.47133694902946522029448389665, −4.27136146150893169210941402139, −3.26959418170819778948295080314, −1.89548471789786727618743036326, 1.82672086631932621475769859904, 3.03001205662580554556629706430, 4.48935313608711231410239835266, 6.38281220615898535866442789030, 6.76498847805699950272269908964, 7.70201623571862959894756901037, 9.145634621641765164632073342620, 9.545705558954617189076775422765, 11.32746135134719920613757504489, 12.25686106710202596782606239477

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