↑

Properties

Label 2-234-117.110-c1-0-3
Degree $2$
Conductor $234$
Sign $0.385 - 0.922i$
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  + (0.258 − 0.965i)2-s + (0.539 + 1.64i)3-s + (−0.866 − 0.499i)4-s + (−0.995 + 3.71i)5-s + (1.72 − 0.0946i)6-s + (−1.54 − 1.54i)7-s + (−0.707 + 0.707i)8-s + (−2.41 + 1.77i)9-s + (3.33 + 1.92i)10-s + (4.04 + 1.08i)11-s + (0.356 − 1.69i)12-s + (−0.109 + 3.60i)13-s + (−1.89 + 1.09i)14-s + (−6.65 + 0.363i)15-s + (0.500 + 0.866i)16-s + (−0.465 − 0.806i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.311 + 0.950i)3-s + (−0.433 − 0.249i)4-s + (−0.445 + 1.66i)5-s + (0.706 − 0.0386i)6-s + (−0.584 − 0.584i)7-s + (−0.249 + 0.249i)8-s + (−0.806 + 0.591i)9-s + (1.05 + 0.607i)10-s + (1.21 + 0.326i)11-s + (0.102 − 0.489i)12-s + (−0.0303 + 0.999i)13-s + (−0.506 + 0.292i)14-s + (−1.71 + 0.0939i)15-s + (0.125 + 0.216i)16-s + (−0.112 − 0.195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.385 - 0.922i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.385 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00811 + 0.671208i\)
\(L(\frac12)\) \(\approx\) \(1.00811 + 0.671208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (-0.539 - 1.64i)T \)
13 \( 1 + (0.109 - 3.60i)T \)
good5 \( 1 + (0.995 - 3.71i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.54 + 1.54i)T + 7iT^{2} \)
11 \( 1 + (-4.04 - 1.08i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.465 + 0.806i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.86 + 0.501i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 7.82T + 23T^{2} \)
29 \( 1 + (-6.41 + 3.70i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.24 - 0.333i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-4.93 + 1.32i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (7.74 + 7.74i)T + 41iT^{2} \)
43 \( 1 + 1.96iT - 43T^{2} \)
47 \( 1 + (-2.52 - 9.40i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 2.04iT - 53T^{2} \)
59 \( 1 + (-1.63 - 6.09i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 0.861T + 61T^{2} \)
67 \( 1 + (-2.36 + 2.36i)T - 67iT^{2} \)
71 \( 1 + (2.90 - 10.8i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.45 + 2.45i)T + 73iT^{2} \)
79 \( 1 + (-6.38 + 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.43 - 1.45i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.522 + 1.95i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (12.4 - 12.4i)T - 97iT^{2} \)
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

−11.95489989653346818819974840015, −11.23273612516570377509575258316, −10.51777891432885080601650554615, −9.732406472304928840717189918424, −8.832160158581060440755108684821, −7.17694657667313056487328441107, −6.39827898181157039605878973006, −4.44027798288675956372816251698, −3.66536973299253738227208653432, −2.64979875031814792212423262774, 0.989714828397604233899786077472, 3.30286117911076661182277341688, 4.82473721503391965537226704992, 5.94282307839985057193348239896, 6.91081941951150968857867192771, 8.310415715815148692957243439973, 8.621431244131149750351889451088, 9.519911684566995856999771884730, 11.52741757899511342874451378671, 12.43967501056726781252612764912

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