↑

Properties

Label 2-234-117.110-c1-0-6
Degree $2$
Conductor $234$
Sign $0.955 + 0.295i$
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 + (1.41 + 0.997i)3-s + (−0.866 − 0.499i)4-s + (0.175 − 0.653i)5-s + (1.33 − 1.10i)6-s + (2.85 + 2.85i)7-s + (−0.707 + 0.707i)8-s + (1.00 + 2.82i)9-s + (−0.585 − 0.338i)10-s + (−0.686 − 0.183i)11-s + (−0.726 − 1.57i)12-s + (−1.25 − 3.37i)13-s + (3.49 − 2.01i)14-s + (0.899 − 0.749i)15-s + (0.500 + 0.866i)16-s + (−3.24 − 5.61i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.817 + 0.576i)3-s + (−0.433 − 0.249i)4-s + (0.0782 − 0.292i)5-s + (0.543 − 0.452i)6-s + (1.07 + 1.07i)7-s + (−0.249 + 0.249i)8-s + (0.336 + 0.941i)9-s + (−0.185 − 0.106i)10-s + (−0.206 − 0.0554i)11-s + (−0.209 − 0.453i)12-s + (−0.348 − 0.937i)13-s + (0.934 − 0.539i)14-s + (0.232 − 0.193i)15-s + (0.125 + 0.216i)16-s + (−0.786 − 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.955 + 0.295i$
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: \((2,\ 234,\ (\ :1/2),\ 0.955 + 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72339 - 0.260457i\)
\(L(\frac12)\) \(\approx\) \(1.72339 - 0.260457i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - 0.997i)T \)
13 \( 1 + (1.25 + 3.37i)T \)
good5 \( 1 + (-0.175 + 0.653i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.85 - 2.85i)T + 7iT^{2} \)
11 \( 1 + (0.686 + 0.183i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.24 + 5.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.253 + 0.0679i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 1.72T + 23T^{2} \)
29 \( 1 + (1.11 - 0.643i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.89 + 1.04i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (7.96 - 2.13i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (6.55 + 6.55i)T + 41iT^{2} \)
43 \( 1 - 6.55iT - 43T^{2} \)
47 \( 1 + (2.43 + 9.07i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 6.34iT - 53T^{2} \)
59 \( 1 + (-1.28 - 4.81i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 3.29T + 61T^{2} \)
67 \( 1 + (7.38 - 7.38i)T - 67iT^{2} \)
71 \( 1 + (-2.09 + 7.82i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.40 + 1.40i)T + 73iT^{2} \)
79 \( 1 + (-7.29 + 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.90 + 1.04i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.64 - 9.85i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.42 + 1.42i)T - 97iT^{2} \)
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

−12.06218718195228584119998363970, −11.15214373701501040602168481528, −10.25938295167954976180027701491, −9.052144421243428095029798329268, −8.654265551674834557687883829852, −7.44566624311209679931670124033, −5.28019681649474610472446996121, −4.85386068646394326526537338509, −3.16116900794682104682756825209, −2.07452963107968649640872375981, 1.82762826177729566007012590123, 3.72884354584657292597283471594, 4.76796265976545632686430333166, 6.52758211138307054264966857519, 7.19689901648923734766706353776, 8.142684083484718137700449080358, 8.908622866513041761111967758373, 10.25098527271833662780056202389, 11.26489596956336822168851792694, 12.55236545890629650285981386558

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