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Properties

Label 2-114-57.50-c3-0-2
Degree $2$
Conductor $114$
Sign $-0.997 - 0.0752i$
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 + (−2.58 − 4.50i)3-s + (−1.99 + 3.46i)4-s + (3.59 − 2.07i)5-s + (5.22 − 8.98i)6-s − 27.3·7-s − 7.99·8-s + (−13.6 + 23.2i)9-s + (7.19 + 4.15i)10-s + 64.4i·11-s + (20.7 + 0.0717i)12-s + (−36.4 − 21.0i)13-s + (−27.3 − 47.3i)14-s + (−18.6 − 10.8i)15-s + (−8 − 13.8i)16-s + (8.65 − 4.99i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.497 − 0.867i)3-s + (−0.249 + 0.433i)4-s + (0.321 − 0.185i)5-s + (0.355 − 0.611i)6-s − 1.47·7-s − 0.353·8-s + (−0.505 + 0.862i)9-s + (0.227 + 0.131i)10-s + 1.76i·11-s + (0.499 + 0.00172i)12-s + (−0.776 − 0.448i)13-s + (−0.521 − 0.903i)14-s + (−0.320 − 0.186i)15-s + (−0.125 − 0.216i)16-s + (0.123 − 0.0713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0752i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.997 - 0.0752i$
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: \((2,\ 114,\ (\ :3/2),\ -0.997 - 0.0752i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00899373 + 0.238673i\)
\(L(\frac12)\) \(\approx\) \(0.00899373 + 0.238673i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.58 + 4.50i)T \)
19 \( 1 + (72.7 + 39.5i)T \)
good5 \( 1 + (-3.59 + 2.07i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + 27.3T + 343T^{2} \)
11 \( 1 - 64.4iT - 1.33e3T^{2} \)
13 \( 1 + (36.4 + 21.0i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-8.65 + 4.99i)T + (2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (114. + 66.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-134. + 232. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 264. iT - 2.97e4T^{2} \)
37 \( 1 + 399. iT - 5.06e4T^{2} \)
41 \( 1 + (-143. - 248. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-125. - 217. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (56.2 + 32.4i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (58.9 - 102. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-13.2 - 22.9i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (127. - 220. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (210. + 121. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (411. + 712. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (219. + 380. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (301. - 174. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 632. iT - 5.71e5T^{2} \)
89 \( 1 + (182. - 316. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (412. - 238. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.35500834404457477590272171622, −12.60878932588727622541189583420, −12.19483992855190282168876763687, −10.31997506859585614356140795200, −9.374356536483464033852968663348, −7.74003705242383453247725398913, −6.82857865244408540899375778016, −5.97468215406271897371816703085, −4.58155058590889809694393643071, −2.45322334049697507182398338755, 0.11306626104939378668865311678, 2.95888316889863693936706593051, 4.02872634101956240576974312127, 5.73676169368745412666742502234, 6.40030737353622915012190255042, 8.696073555862639791950793912902, 9.808455081656187147591363178020, 10.38669350037251515503314714565, 11.53784237492229066425863169241, 12.45069503096884461105019640633

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