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Properties

Label 2-114-19.11-c3-0-8
Degree $2$
Conductor $114$
Sign $-0.910 - 0.412i$
Analytic cond. $6.72621$
Root an. cond. $2.59349$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

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 + (−1 + 1.73i)5-s + (3 + 5.19i)6-s − 21·7-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (1.99 + 3.46i)10-s − 40·11-s + 12·12-s + (−8.5 − 14.7i)13-s + (−21 + 36.3i)14-s + (−3 − 5.19i)15-s + (−8 + 13.8i)16-s + (−18 + 31.1i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.0894 + 0.154i)5-s + (0.204 + 0.353i)6-s − 1.13·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.0632 + 0.109i)10-s − 1.09·11-s + 0.288·12-s + (−0.181 − 0.314i)13-s + (−0.400 + 0.694i)14-s + (−0.0516 − 0.0894i)15-s + (−0.125 + 0.216i)16-s + (−0.256 + 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.910 - 0.412i$
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: \(1\)
Selberg data: \((2,\ 114,\ (\ :3/2),\ -0.910 - 0.412i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + (76 - 32.9i)T \)
good5 \( 1 + (1 - 1.73i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + 21T + 343T^{2} \)
11 \( 1 + 40T + 1.33e3T^{2} \)
13 \( 1 + (8.5 + 14.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (18 - 31.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (37 + 64.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (50 + 86.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 103T + 2.97e4T^{2} \)
37 \( 1 - 187T + 5.06e4T^{2} \)
41 \( 1 + (-64 + 110. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (60.5 - 104. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (205 + 355. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (115 + 199. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-372 + 644. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-138.5 - 239. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-115.5 - 200. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (289 - 500. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (304.5 - 527. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (629.5 - 1.09e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 696T + 5.71e5T^{2} \)
89 \( 1 + (-306 - 530. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-775 + 1.34e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.73916650378305619495854572218, −11.38123570463625269226395067414, −10.38358204948418153651095580219, −9.765108478447073019015467603586, −8.332502146744416987728103523825, −6.60166823859674941996319659563, −5.44064854089498148039607764008, −4.00004182325913001412303918894, −2.69008936743913888765634550373, 0, 2.79763033543665769729711732584, 4.61257630574850714318613260702, 5.98090909886034794711634773996, 6.90912872215452073318239733632, 8.028796222208092094559783516480, 9.304550973083164130757031955570, 10.57856798386827375614239409340, 11.94697761180547833740940030668, 12.94741871954418398059413206255

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