↑

Properties

Label 2-234-117.103-c1-0-6
Degree $2$
Conductor $234$
Sign $0.998 + 0.0613i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (1.21 − 1.23i)3-s + (0.499 − 0.866i)4-s + (2.73 + 1.57i)5-s + (−0.434 + 1.67i)6-s + (1.36 − 0.787i)7-s + 0.999i·8-s + (−0.0487 − 2.99i)9-s − 3.15·10-s + (−3.26 + 1.88i)11-s + (−0.461 − 1.66i)12-s + (−3.47 + 0.966i)13-s + (−0.787 + 1.36i)14-s + (5.27 − 1.45i)15-s + (−0.5 − 0.866i)16-s + 7.06·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.701 − 0.712i)3-s + (0.249 − 0.433i)4-s + (1.22 + 0.706i)5-s + (−0.177 + 0.684i)6-s + (0.515 − 0.297i)7-s + 0.353i·8-s + (−0.0162 − 0.999i)9-s − 0.998·10-s + (−0.983 + 0.567i)11-s + (−0.133 − 0.481i)12-s + (−0.963 + 0.268i)13-s + (−0.210 + 0.364i)14-s + (1.36 − 0.376i)15-s + (−0.125 − 0.216i)16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.998 + 0.0613i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.998 + 0.0613i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36024 - 0.0417435i\)
\(L(\frac12)\) \(\approx\) \(1.36024 - 0.0417435i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-1.21 + 1.23i)T \)
13 \( 1 + (3.47 - 0.966i)T \)
good5 \( 1 + (-2.73 - 1.57i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.36 + 0.787i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.26 - 1.88i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 7.06T + 17T^{2} \)
19 \( 1 + 3.76iT - 19T^{2} \)
23 \( 1 + (1.84 - 3.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.109 - 0.189i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.65 + 1.53i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.292iT - 37T^{2} \)
41 \( 1 + (-6.39 - 3.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.05 + 5.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.17 - 3.56i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 14.4T + 53T^{2} \)
59 \( 1 + (9.04 + 5.22i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.00 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.33 - 3.65i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.772iT - 71T^{2} \)
73 \( 1 + 13.5iT - 73T^{2} \)
79 \( 1 + (-6.34 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.314 + 0.181i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.06iT - 89T^{2} \)
97 \( 1 + (-0.535 + 0.309i)T + (48.5 - 84.0i)T^{2} \)
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

−12.28900601578666752155789182507, −10.96518797330657632816068457512, −9.840878691215291256760115801013, −9.491150359066303318437397334111, −7.888627196093042073480478441155, −7.43657083650313929678038110742, −6.33671845116051119073235757062, −5.16532424263268325667529000304, −2.85889942830648546235385732899, −1.76409765835332206222012913033, 1.87602708748577297051869169750, 3.11372949108653322685213205989, 4.93881954450274170371056278386, 5.71739817690431645319953636788, 7.81554473651248109205300145001, 8.333273120087606100929493044489, 9.531785293306634158287182261048, 9.973303046699107096779236769654, 10.82515767816394440865436915401, 12.26278363757293837890398884475

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