↑

Properties

Label 2-114-19.11-c3-0-9
Degree $2$
Conductor $114$
Sign $-0.827 + 0.561i$
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 + (6.49 − 11.2i)5-s + (−3 − 5.19i)6-s − 25.6·7-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (−12.9 − 22.4i)10-s + 26.7·11-s − 12·12-s + (4.29 + 7.44i)13-s + (−25.6 + 44.3i)14-s + (−19.4 − 33.7i)15-s + (−8 + 13.8i)16-s + (−4.08 + 7.07i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.580 − 1.00i)5-s + (−0.204 − 0.353i)6-s − 1.38·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.410 − 0.711i)10-s + 0.733·11-s − 0.288·12-s + (0.0916 + 0.158i)13-s + (−0.488 + 0.846i)14-s + (−0.335 − 0.580i)15-s + (−0.125 + 0.216i)16-s + (−0.0582 + 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.827 + 0.561i$
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: \((2,\ 114,\ (\ :3/2),\ -0.827 + 0.561i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.532481 - 1.73124i\)
\(L(\frac12)\) \(\approx\) \(0.532481 - 1.73124i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.5 + 82.0i)T \)
good5 \( 1 + (-6.49 + 11.2i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + 25.6T + 343T^{2} \)
11 \( 1 - 26.7T + 1.33e3T^{2} \)
13 \( 1 + (-4.29 - 7.44i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (4.08 - 7.07i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (78.4 + 135. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-103. - 179. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 94.5T + 2.97e4T^{2} \)
37 \( 1 - 197.T + 5.06e4T^{2} \)
41 \( 1 + (-188. + 326. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-254. + 440. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-183. - 317. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-101. - 176. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (296. - 512. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-254. - 440. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-125. - 217. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-57.7 + 99.9i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-416. + 721. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (184. - 319. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.28e3T + 5.71e5T^{2} \)
89 \( 1 + (30.5 + 52.8i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (123. - 214. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−12.65470033491529389563489348589, −12.13693139225649364123340775860, −10.53922282060538139175888186358, −9.324951370095771417462110128168, −8.825076314668147308865705996844, −6.85915770406683865314038358087, −5.81889782441868448295012242101, −4.21889531714895978034052843759, −2.62445552557371477156189576065, −0.878399618539942468101899467964, 2.86134946042847222791301606996, 3.96549667906809138790946506277, 5.93387226490718014826913399535, 6.55365354972886238446239958690, 7.947543011821978595645564906788, 9.584459845635276879201203751970, 9.942611521436596092014211811097, 11.45905653610382522124391330243, 12.78553169309075877067117679515, 13.82462681588004425654714936284

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