↑

Properties

Label 2-117-13.3-c3-0-16
Degree $2$
Conductor $117$
Sign $-0.872 + 0.488i$
Analytic cond. $6.90322$
Root an. cond. $2.62739$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (3.5 − 6.06i)4-s − 7·5-s + (5 − 8.66i)7-s − 15·8-s + (3.5 + 6.06i)10-s + (−11 − 19.0i)11-s + (−45.5 + 11.2i)13-s − 10·14-s + (−20.5 − 35.5i)16-s + (18.5 − 32.0i)17-s + (−15 + 25.9i)19-s + (−24.5 + 42.4i)20-s + (−11 + 19.0i)22-s + (−81 − 140. i)23-s + ⋯
L(s)  = 1  + (−0.176 − 0.306i)2-s + (0.437 − 0.757i)4-s − 0.626·5-s + (0.269 − 0.467i)7-s − 0.662·8-s + (0.110 + 0.191i)10-s + (−0.301 − 0.522i)11-s + (−0.970 + 0.240i)13-s − 0.190·14-s + (−0.320 − 0.554i)16-s + (0.263 − 0.457i)17-s + (−0.181 + 0.313i)19-s + (−0.273 + 0.474i)20-s + (−0.106 + 0.184i)22-s + (−0.734 − 1.27i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.242476 - 0.928711i\)
\(L(\frac12)\) \(\approx\) \(0.242476 - 0.928711i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (45.5 - 11.2i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + 7T + 125T^{2} \)
7 \( 1 + (-5 + 8.66i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (11 + 19.0i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-18.5 + 32.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (15 - 25.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (81 + 140. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (56.5 + 97.8i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 196T + 2.97e4T^{2} \)
37 \( 1 + (6.5 + 11.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-142.5 - 246. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-123 + 213. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 462T + 1.03e5T^{2} \)
53 \( 1 - 537T + 1.48e5T^{2} \)
59 \( 1 + (-288 + 498. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-317.5 + 549. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (101 + 174. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (543 - 940. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 805T + 3.89e5T^{2} \)
79 \( 1 - 884T + 4.93e5T^{2} \)
83 \( 1 + 518T + 5.71e5T^{2} \)
89 \( 1 + (-97 - 168. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-601 + 1.04e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.28231910944534804119083051430, −11.53839096003620194927751393593, −10.53415490943676177986605910275, −9.703606639102658765345659685289, −8.235763759568026355939236712767, −7.12309528220031343746535170811, −5.78957357639206751218916549411, −4.32093520581000267638792085711, −2.47157683223253399062240804461, −0.51489570021404940721126525215, 2.45502750792839001416356804537, 4.01891154616426682799978641321, 5.65252689261565863620455886015, 7.24140063109086046178463494018, 7.85484771614457677708122188164, 8.996118664884651228692808733173, 10.36069691942410534265700567565, 11.80139406681030646107256935727, 12.13483456745135521715453710506, 13.33100090295058168242163435457

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