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Properties

Label 2-114-19.11-c3-0-2
Degree $2$
Conductor $114$
Sign $-0.831 - 0.556i$
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 + (1.5 − 2.59i)3-s + (−1.99 − 3.46i)4-s + (−7.12 + 12.3i)5-s + (3 + 5.19i)6-s + 13.2·7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (−14.2 − 24.6i)10-s − 65.9·11-s − 12·12-s + (34.2 + 59.3i)13-s + (−13.2 + 22.9i)14-s + (21.3 + 37.0i)15-s + (−8 + 13.8i)16-s + (−49.8 + 86.3i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.637 + 1.10i)5-s + (0.204 + 0.353i)6-s + 0.715·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.450 − 0.780i)10-s − 1.80·11-s − 0.288·12-s + (0.730 + 1.26i)13-s + (−0.253 + 0.438i)14-s + (0.368 + 0.637i)15-s + (−0.125 + 0.216i)16-s + (−0.711 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.556i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.831 - 0.556i$
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: \((2,\ 114,\ (\ :3/2),\ -0.831 - 0.556i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.234098 + 0.770635i\)
\(L(\frac12)\) \(\approx\) \(0.234098 + 0.770635i\)
\(L(\frac{5}{2})\) 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 \( 1 + (80.3 - 20.0i)T \)
good5 \( 1 + (7.12 - 12.3i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 - 13.2T + 343T^{2} \)
11 \( 1 + 65.9T + 1.33e3T^{2} \)
13 \( 1 + (-34.2 - 59.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (49.8 - 86.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (1.76 + 3.06i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-40.9 - 70.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 247.T + 2.97e4T^{2} \)
37 \( 1 - 421.T + 5.06e4T^{2} \)
41 \( 1 + (-172. + 299. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-183. + 317. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-45.4 - 78.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-344. - 596. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (91.1 - 157. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-0.258 - 0.448i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-79.9 - 138. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-395. + 685. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (161. - 278. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (159. - 275. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 684.T + 5.71e5T^{2} \)
89 \( 1 + (360. + 623. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (207. - 359. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.73311612153167891160543266948, −12.66211332092279585619815190278, −11.00689438097596345154187261273, −10.71821349268241654620934483631, −8.849770300136172302017203796663, −7.937049328061202311180743869282, −7.12506609299480865895281578604, −5.95217482979652277168261672740, −4.12468586059776912343222214405, −2.18951877015826184875813969342, 0.45759803224516073293198667891, 2.66156462675325810189319042848, 4.38350322644403943800679331087, 5.27806520592706187380906568668, 7.908637952028679568258315514336, 8.247660108248154765040275760863, 9.450057198756971560835653719895, 10.74133985695155729076964837464, 11.35613622851590139246566303007, 12.90054452627768691898700685791

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