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Properties

Label 2-114-57.50-c3-0-4
Degree $2$
Conductor $114$
Sign $-0.690 + 0.723i$
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 + (−2.74 + 4.41i)3-s + (−1.99 + 3.46i)4-s + (−1.43 + 0.830i)5-s + (−10.3 − 0.332i)6-s − 19.8·7-s − 7.99·8-s + (−11.9 − 24.1i)9-s + (−2.87 − 1.66i)10-s − 21.2i·11-s + (−9.81 − 18.3i)12-s + (48.9 + 28.2i)13-s + (−19.8 − 34.3i)14-s + (0.276 − 8.62i)15-s + (−8 − 13.8i)16-s + (−105. + 60.7i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.527 + 0.849i)3-s + (−0.249 + 0.433i)4-s + (−0.128 + 0.0743i)5-s + (−0.706 − 0.0226i)6-s − 1.07·7-s − 0.353·8-s + (−0.443 − 0.896i)9-s + (−0.0910 − 0.0525i)10-s − 0.583i·11-s + (−0.236 − 0.440i)12-s + (1.04 + 0.603i)13-s + (−0.378 − 0.656i)14-s + (0.00475 − 0.148i)15-s + (−0.125 − 0.216i)16-s + (−1.50 + 0.867i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.690 + 0.723i$
Analytic conductor: \(6.72621\)
Root analytic conductor: \(2.59349\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :3/2),\ -0.690 + 0.723i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.184617 - 0.431581i\)
\(L(\frac12)\) \(\approx\) \(0.184617 - 0.431581i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.74 - 4.41i)T \)
19 \( 1 + (-8.99 + 82.3i)T \)
good5 \( 1 + (1.43 - 0.830i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + 19.8T + 343T^{2} \)
11 \( 1 + 21.2iT - 1.33e3T^{2} \)
13 \( 1 + (-48.9 - 28.2i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (105. - 60.7i)T + (2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (136. + 78.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (54.8 - 94.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 44.3iT - 2.97e4T^{2} \)
37 \( 1 - 343. iT - 5.06e4T^{2} \)
41 \( 1 + (40.7 + 70.5i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-206. - 357. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (152. + 88.1i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (152. - 263. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-135. - 233. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-104. + 181. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (479. + 276. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-220. - 382. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-141. - 245. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-929. + 536. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.16e3iT - 5.71e5T^{2} \)
89 \( 1 + (32.4 - 56.1i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-687. + 396. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.72850157585382749633400804415, −12.92065578770143087714646428098, −11.58452773023477154276924548387, −10.71876813665266000916234239001, −9.383015379926114836260154349284, −8.546873429845741244025849251822, −6.61562637779931216755237178718, −6.06424362210160949345527791480, −4.46208583767553545051900333560, −3.39284101016946407098286259296, 0.22937811862313039883441595706, 2.17563190526877716311198427568, 3.91736623176203576685421895407, 5.66252564083136200374302951039, 6.58025159930954433051115518344, 7.962117515861724685261335083322, 9.447335695681642978724234411905, 10.57642512869375780304597798981, 11.62993693164464147092200574027, 12.49043050614185348296066500551

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