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Properties

Label 2-114-57.8-c3-0-6
Degree $2$
Conductor $114$
Sign $0.830 - 0.556i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (1.03 + 5.09i)3-s + (−1.99 − 3.46i)4-s + (9.82 + 5.67i)5-s + (9.85 + 3.30i)6-s + 5.23·7-s − 7.99·8-s + (−24.8 + 10.5i)9-s + (19.6 − 11.3i)10-s + 50.0i·11-s + (15.5 − 13.7i)12-s + (22.4 − 12.9i)13-s + (5.23 − 9.07i)14-s + (−18.7 + 55.8i)15-s + (−8 + 13.8i)16-s + (70.4 + 40.6i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.198 + 0.980i)3-s + (−0.249 − 0.433i)4-s + (0.878 + 0.507i)5-s + (0.670 + 0.224i)6-s + 0.282·7-s − 0.353·8-s + (−0.921 + 0.389i)9-s + (0.621 − 0.358i)10-s + 1.37i·11-s + (0.374 − 0.331i)12-s + (0.479 − 0.276i)13-s + (0.100 − 0.173i)14-s + (−0.322 + 0.962i)15-s + (−0.125 + 0.216i)16-s + (1.00 + 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.830 - 0.556i$
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: \((2,\ 114,\ (\ :3/2),\ 0.830 - 0.556i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.14550 + 0.652587i\)
\(L(\frac12)\) \(\approx\) \(2.14550 + 0.652587i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.03 - 5.09i)T \)
19 \( 1 + (-79.2 + 23.8i)T \)
good5 \( 1 + (-9.82 - 5.67i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 - 5.23T + 343T^{2} \)
11 \( 1 - 50.0iT - 1.33e3T^{2} \)
13 \( 1 + (-22.4 + 12.9i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-70.4 - 40.6i)T + (2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (44.6 - 25.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (100. + 174. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 49.0iT - 2.97e4T^{2} \)
37 \( 1 - 154. iT - 5.06e4T^{2} \)
41 \( 1 + (-175. + 303. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (3.51 - 6.08i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (167. - 96.6i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (152. + 263. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-299. + 518. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (285. + 494. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (354. - 204. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (224. - 389. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-520. + 901. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-933. - 538. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 656. iT - 5.71e5T^{2} \)
89 \( 1 + (171. + 297. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.40e3 - 810. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.34756774718022749113452116627, −12.05152009632742598144159494891, −10.95553949956547362328123979634, −9.945656705070020029317616729117, −9.563125423145683371358105846266, −7.891271295609285366397568150432, −6.04795107571413895144340108826, −4.94760406551271457420060030393, −3.56269219773955016557422330234, −2.08121663738766325815107849139, 1.24291530244636335695893365812, 3.22461337588290316117670053947, 5.39898681831888138769788217648, 6.07953541728928712374889415874, 7.46628758711841045686414217725, 8.489910234567410940614990250182, 9.425679959155106436399801336145, 11.21567342159561402031191867262, 12.26795445864381566697719976285, 13.32184518986044007672458578471

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