Properties

Label 2-114-19.7-c3-0-4
Degree $2$
Conductor $114$
Sign $0.980 + 0.194i$
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 + (−2.09 − 3.62i)5-s + (3 − 5.19i)6-s + 3.18·7-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (−4.18 + 7.25i)10-s + 69.4·11-s − 12·12-s + (4.06 − 7.03i)13-s + (−3.18 − 5.52i)14-s + (6.28 − 10.8i)15-s + (−8 − 13.8i)16-s + (53.0 + 91.8i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.187 − 0.324i)5-s + (0.204 − 0.353i)6-s + 0.172·7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.132 + 0.229i)10-s + 1.90·11-s − 0.288·12-s + (0.0867 − 0.150i)13-s + (−0.0608 − 0.105i)14-s + (0.108 − 0.187i)15-s + (−0.125 − 0.216i)16-s + (0.756 + 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.980 + 0.194i$
Analytic conductor: \(6.72621\)
Root analytic conductor: \(2.59349\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :3/2),\ 0.980 + 0.194i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.53546 - 0.150378i\)
\(L(\frac12)\) \(\approx\) \(1.53546 - 0.150378i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-42.6 - 70.9i)T \)
good5 \( 1 + (2.09 + 3.62i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 - 3.18T + 343T^{2} \)
11 \( 1 - 69.4T + 1.33e3T^{2} \)
13 \( 1 + (-4.06 + 7.03i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-53.0 - 91.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-88.2 + 152. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-33.1 + 57.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 140.T + 2.97e4T^{2} \)
37 \( 1 + 156.T + 5.06e4T^{2} \)
41 \( 1 + (-207. - 359. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (57.9 + 100. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (310. - 537. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-185. + 322. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (45.8 + 79.4i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (109. - 189. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-72.6 + 125. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (443. + 768. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (99.5 + 172. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-194. - 337. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 380.T + 5.71e5T^{2} \)
89 \( 1 + (-212. + 368. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (209. + 363. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.77262728552376000545287406269, −12.00149682464457135983482004236, −10.91510371970120811818079117607, −9.870983808653618308847078356911, −8.891040319746879435841632483113, −8.039343626673847500498304025331, −6.32425332911062646120108327208, −4.50155438833076379654555140690, −3.43950171241014554367180577565, −1.37244729643852202861215760880, 1.23161102649562794158986510908, 3.46670369489069848524575477029, 5.29695464713185838189864783887, 6.86938493680807970628693256970, 7.33979911940053856171184913567, 8.924256945638157890260441164838, 9.482463430252967589015214696838, 11.22148550212298954619264545886, 11.95490428546494127058363217148, 13.45683425558209076638662135083

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