↑

Properties

Label 2-114-57.8-c3-0-16
Degree $2$
Conductor $114$
Sign $-0.987 - 0.157i$
Analytic cond. $6.72621$
Root an. cond. $2.59349$
Motivic weight $3$
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 − 1.73i)2-s + (2.25 + 4.67i)3-s + (−1.99 − 3.46i)4-s + (−17.8 − 10.3i)5-s + (10.3 + 0.766i)6-s − 12.1·7-s − 7.99·8-s + (−16.7 + 21.1i)9-s + (−35.7 + 20.6i)10-s + 21.4i·11-s + (11.6 − 17.1i)12-s + (−3.73 + 2.15i)13-s + (−12.1 + 21.0i)14-s + (7.92 − 107. i)15-s + (−8 + 13.8i)16-s + (−100. − 57.9i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.434 + 0.900i)3-s + (−0.249 − 0.433i)4-s + (−1.60 − 0.923i)5-s + (0.705 + 0.0521i)6-s − 0.655·7-s − 0.353·8-s + (−0.622 + 0.782i)9-s + (−1.13 + 0.653i)10-s + 0.587i·11-s + (0.281 − 0.413i)12-s + (−0.0796 + 0.0459i)13-s + (−0.231 + 0.401i)14-s + (0.136 − 1.84i)15-s + (−0.125 + 0.216i)16-s + (−1.43 − 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.987 - 0.157i$
Analytic conductor: \(6.72621\)
Root analytic conductor: \(2.59349\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :3/2),\ -0.987 - 0.157i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0167032 + 0.211012i\)
\(L(\frac12)\) \(\approx\) \(0.0167032 + 0.211012i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 + (-2.25 - 4.67i)T \)
19 \( 1 + (63.0 + 53.7i)T \)
good5 \( 1 + (17.8 + 10.3i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + 12.1T + 343T^{2} \)
11 \( 1 - 21.4iT - 1.33e3T^{2} \)
13 \( 1 + (3.73 - 2.15i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (100. + 57.9i)T + (2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-157. + 90.9i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-26.1 - 45.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 63.1iT - 2.97e4T^{2} \)
37 \( 1 + 103. iT - 5.06e4T^{2} \)
41 \( 1 + (130. - 226. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-146. + 253. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-62.8 + 36.2i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-261. - 453. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-27.8 + 48.2i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (398. + 690. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (799. - 461. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (388. - 673. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-113. + 196. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (430. + 248. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 407. iT - 5.71e5T^{2} \)
89 \( 1 + (435. + 754. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (23.9 + 13.8i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−12.56374612603269475819454862975, −11.47866569559980491669488625802, −10.66306669989347083508221461189, −9.177611894647876113021647255205, −8.675262453633741277094016100410, −7.09370005881151128210919283243, −4.82233773508900971931296435021, −4.29888747656000267711460486600, −2.91994661848293985417487307837, −0.092803406045943900070052304923, 2.98130064172688215775378439232, 4.00224905167850868186032183229, 6.26534300688404229250725562125, 7.00931197309610607129015683100, 7.973344427644559363347541241674, 8.852314367433175781453129689701, 10.83050228112279250767974377945, 11.76527304693120443336807500558, 12.79165161120193040142648401935, 13.61200766339118499768630337686

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