Properties

Label 2-114-19.11-c3-0-7
Degree $2$
Conductor $114$
Sign $-0.293 + 0.956i$
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 + (1.5 − 2.59i)3-s + (−1.99 − 3.46i)4-s + (2.60 − 4.51i)5-s + (−3 − 5.19i)6-s + 31.4·7-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (−5.21 − 9.03i)10-s − 21.2·11-s − 12·12-s + (−28.1 − 48.7i)13-s + (31.4 − 54.5i)14-s + (−7.82 − 13.5i)15-s + (−8 + 13.8i)16-s + (−8.63 + 14.9i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.233 − 0.403i)5-s + (−0.204 − 0.353i)6-s + 1.70·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.164 − 0.285i)10-s − 0.581·11-s − 0.288·12-s + (−0.600 − 1.03i)13-s + (0.601 − 1.04i)14-s + (−0.134 − 0.233i)15-s + (−0.125 + 0.216i)16-s + (−0.123 + 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.293 + 0.956i$
Analytic conductor: \(6.72621\)
Root analytic conductor: \(2.59349\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :3/2),\ -0.293 + 0.956i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.34093 - 1.81400i\)
\(L(\frac12)\) \(\approx\) \(1.34093 - 1.81400i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.5 + 2.59i)T \)
19 \( 1 + (42.3 + 71.1i)T \)
good5 \( 1 + (-2.60 + 4.51i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 - 31.4T + 343T^{2} \)
11 \( 1 + 21.2T + 1.33e3T^{2} \)
13 \( 1 + (28.1 + 48.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (8.63 - 14.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (-103. - 178. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (103. + 178. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 129.T + 2.97e4T^{2} \)
37 \( 1 - 440.T + 5.06e4T^{2} \)
41 \( 1 + (217. - 377. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (64.9 - 112. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (54.2 + 93.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-203. - 352. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-58.1 + 100. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-170. - 295. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-105. - 182. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (79.0 - 136. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (286. - 496. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-442. + 766. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 573.T + 5.71e5T^{2} \)
89 \( 1 + (107. + 186. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (264. - 458. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−13.07803282938288201314455347598, −11.67422332470828514561911468501, −11.03501406955039054715347403679, −9.665546762872257128863950138633, −8.376848457823580912551053322375, −7.52238814580754908277602611803, −5.56109817647016539174293206730, −4.64775021617587382419607028476, −2.64737401194790779933805844914, −1.22085570205281891918273391839, 2.34122756250011474824872651977, 4.34444498840724671519273008958, 5.16260465009843706075090520212, 6.77670778135530802852219123197, 8.001514799953768055559810581762, 8.850193603111071504216697307292, 10.34602765168322496945529341407, 11.26022411901298016836385759527, 12.46121108607220419180430720029, 13.86849740703594190185599124179

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