↑

Properties

Label 2-117-117.88-c1-0-4
Degree $2$
Conductor $117$
Sign $0.868 - 0.496i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.391i·2-s + (0.608 + 1.62i)3-s + 1.84·4-s + (−1.60 + 0.927i)5-s + (0.635 − 0.238i)6-s + (−0.0712 + 0.0411i)7-s − 1.50i·8-s + (−2.25 + 1.97i)9-s + (0.363 + 0.629i)10-s − 0.0536i·11-s + (1.12 + 2.99i)12-s + (1.83 − 3.10i)13-s + (0.0161 + 0.0279i)14-s + (−2.48 − 2.04i)15-s + 3.10·16-s + (2.16 − 3.75i)17-s + ⋯
L(s)  = 1  − 0.276i·2-s + (0.351 + 0.936i)3-s + 0.923·4-s + (−0.718 + 0.414i)5-s + (0.259 − 0.0972i)6-s + (−0.0269 + 0.0155i)7-s − 0.532i·8-s + (−0.753 + 0.657i)9-s + (0.114 + 0.198i)10-s − 0.0161i·11-s + (0.324 + 0.864i)12-s + (0.509 − 0.860i)13-s + (0.00430 + 0.00745i)14-s + (−0.640 − 0.526i)15-s + 0.775·16-s + (0.525 − 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.868 - 0.496i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.868 - 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18745 + 0.315607i\)
\(L(\frac12)\) \(\approx\) \(1.18745 + 0.315607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.608 - 1.62i)T \)
13 \( 1 + (-1.83 + 3.10i)T \)
good2 \( 1 + 0.391iT - 2T^{2} \)
5 \( 1 + (1.60 - 0.927i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.0712 - 0.0411i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.0536iT - 11T^{2} \)
17 \( 1 + (-2.16 + 3.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.45 + 2.57i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.790 - 1.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.21T + 29T^{2} \)
31 \( 1 + (0.647 - 0.374i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.767 + 0.443i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.0 + 5.83i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.89 - 5.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.48 - 4.89i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 - 6.02iT - 59T^{2} \)
61 \( 1 + (5.44 + 9.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.76 + 2.17i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.1 - 6.45i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.76iT - 73T^{2} \)
79 \( 1 + (5.17 - 8.96i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.68 - 5.59i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.76 - 5.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.66 - 5.57i)T + (48.5 - 84.0i)T^{2} \)
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

−13.72719655983180727482782817637, −12.35223006354591271636149583052, −11.21356516575655977554244149891, −10.74697664463634927188772784754, −9.600191486494140350948751482346, −8.211914107641036922414512791124, −7.16505763259475699888847722293, −5.60174002455161016887943288654, −3.85763211870559412056778111744, −2.78591249598120888039081054565, 1.90112242617466206128221893347, 3.77715598655505776145921808456, 5.92437348239759544092984083053, 6.87691640294734540609465047120, 7.980520853055108667360407789071, 8.674434116538542322970555554966, 10.50236138675978712536472687073, 11.74856609061494271994587271416, 12.24662439469196943617234106578, 13.38934572174651738895811406768

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