↑

Properties

Label 2-234-117.20-c1-0-0
Degree $2$
Conductor $234$
Sign $-0.196 - 0.980i$
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  + (−0.965 − 0.258i)2-s + (−0.679 − 1.59i)3-s + (0.866 + 0.499i)4-s + (−1.16 − 0.313i)5-s + (0.244 + 1.71i)6-s + (−3.18 + 3.18i)7-s + (−0.707 − 0.707i)8-s + (−2.07 + 2.16i)9-s + (1.04 + 0.604i)10-s + (−0.226 + 0.844i)11-s + (0.208 − 1.71i)12-s + (3.53 + 0.709i)13-s + (3.90 − 2.25i)14-s + (0.295 + 2.07i)15-s + (0.500 + 0.866i)16-s + (−1.49 − 2.58i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.392 − 0.919i)3-s + (0.433 + 0.249i)4-s + (−0.522 − 0.140i)5-s + (0.0996 + 0.700i)6-s + (−1.20 + 1.20i)7-s + (−0.249 − 0.249i)8-s + (−0.692 + 0.721i)9-s + (0.331 + 0.191i)10-s + (−0.0681 + 0.254i)11-s + (0.0600 − 0.496i)12-s + (0.980 + 0.196i)13-s + (1.04 − 0.601i)14-s + (0.0762 + 0.535i)15-s + (0.125 + 0.216i)16-s + (−0.362 − 0.627i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.196 - 0.980i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.196 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.157376 + 0.192031i\)
\(L(\frac12)\) \(\approx\) \(0.157376 + 0.192031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (0.679 + 1.59i)T \)
13 \( 1 + (-3.53 - 0.709i)T \)
good5 \( 1 + (1.16 + 0.313i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (3.18 - 3.18i)T - 7iT^{2} \)
11 \( 1 + (0.226 - 0.844i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.49 + 2.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.92 - 7.17i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 0.154T + 23T^{2} \)
29 \( 1 + (8.13 - 4.69i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.13 - 4.22i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.18 + 4.40i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.99 + 3.99i)T - 41iT^{2} \)
43 \( 1 + 2.32iT - 43T^{2} \)
47 \( 1 + (6.31 - 1.69i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 0.732iT - 53T^{2} \)
59 \( 1 + (13.8 - 3.72i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 3.28T + 61T^{2} \)
67 \( 1 + (3.76 + 3.76i)T + 67iT^{2} \)
71 \( 1 + (-14.6 - 3.93i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.594 + 0.594i)T - 73iT^{2} \)
79 \( 1 + (-1.78 + 3.09i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.654 + 2.44i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (5.91 - 1.58i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.88 + 7.88i)T + 97iT^{2} \)
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

−12.42912185860841945393665203697, −11.61911587904902970548305531693, −10.64316393633199789432846392585, −9.343955205207747273687549590840, −8.570312803152238584104599167546, −7.54553568408785884508744819902, −6.43787205022635639327838580695, −5.65863436978248231863240791564, −3.48912642073540497181206815658, −1.97945678319321825212780515714, 0.25129159975907292867187361872, 3.32612749905230765786412331085, 4.24926765228020640123709021008, 5.98891409011739199536437471214, 6.79665798038060028067679617920, 8.024783381718990779682067403885, 9.231338822878937219594960888488, 9.913674104211266992341676507299, 11.07910122711786730753754645892, 11.17787521545850648946508520322

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