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Properties

Label 2-114-19.11-c3-0-1
Degree $2$
Conductor $114$
Sign $-0.364 + 0.931i$
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 + (−7.47 + 12.9i)5-s + (−3 − 5.19i)6-s − 1.95·7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (−14.9 − 25.9i)10-s − 0.954·11-s + 12·12-s + (−24.4 − 42.2i)13-s + (1.95 − 3.38i)14-s + (−22.4 − 38.8i)15-s + (−8 + 13.8i)16-s + (−6 + 10.3i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.668 + 1.15i)5-s + (−0.204 − 0.353i)6-s − 0.105·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.472 − 0.819i)10-s − 0.0261·11-s + 0.288·12-s + (−0.520 − 0.901i)13-s + (0.0373 − 0.0646i)14-s + (−0.386 − 0.668i)15-s + (−0.125 + 0.216i)16-s + (−0.0856 + 0.148i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.364 + 0.931i$
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: \((2,\ 114,\ (\ :3/2),\ -0.364 + 0.931i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0765791 - 0.112206i\)
\(L(\frac12)\) \(\approx\) \(0.0765791 - 0.112206i\)
\(L(\frac{5}{2})\) 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 \( 1 + (36.8 + 74.1i)T \)
good5 \( 1 + (7.47 - 12.9i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + 1.95T + 343T^{2} \)
11 \( 1 + 0.954T + 1.33e3T^{2} \)
13 \( 1 + (24.4 + 42.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (6 - 10.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (25.4 + 44.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (26.9 + 46.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 11.4T + 2.97e4T^{2} \)
37 \( 1 + 176.T + 5.06e4T^{2} \)
41 \( 1 + (126. - 219. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (244. - 423. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-175. - 304. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-92.4 - 160. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (140. - 243. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (281. + 488. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-94.2 - 163. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (53.2 - 92.1i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-491. + 851. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (402. - 696. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.04e3T + 5.71e5T^{2} \)
89 \( 1 + (436. + 756. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (61.4 - 106. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−14.21094889472662156267455346099, −12.79773278801788833600876357322, −11.40400053051179019074621385203, −10.62906642352175204986133211538, −9.700491581103217664004586125737, −8.274564652335883242984978820626, −7.20410024526772535637098731865, −6.17795449449486805686549076196, −4.66926030723518130673697856832, −3.07074942094813570009131334316, 0.083849336909512863229023824573, 1.75737904508269653157245272937, 3.92139843528388033715301749766, 5.22057990393825203778336904607, 7.00550698654492706744932185445, 8.205030494884371724329535182567, 9.052904658701444145559051562529, 10.32510806563894141076763904972, 11.71184860315393602110670309649, 12.16679598420530775764760089514

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