↑

Properties

Label 2-115-23.22-c2-0-13
Degree $2$
Conductor $115$
Sign $0.999 + 0.0321i$
Analytic cond. $3.13352$
Root an. cond. $1.77017$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.24·2-s + 3.98·3-s + 1.05·4-s − 2.23i·5-s + 8.96·6-s + 6.68i·7-s − 6.62·8-s + 6.91·9-s − 5.02i·10-s − 10.4i·11-s + 4.20·12-s − 4.43·13-s + 15.0i·14-s − 8.92i·15-s − 19.1·16-s − 1.02i·17-s + ⋯
L(s)  = 1  + 1.12·2-s + 1.32·3-s + 0.263·4-s − 0.447i·5-s + 1.49·6-s + 0.954i·7-s − 0.827·8-s + 0.768·9-s − 0.502i·10-s − 0.952i·11-s + 0.350·12-s − 0.340·13-s + 1.07i·14-s − 0.594i·15-s − 1.19·16-s − 0.0605i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0321i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.999 + 0.0321i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ 0.999 + 0.0321i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.94140 - 0.0473283i\)
\(L(\frac12)\) \(\approx\) \(2.94140 - 0.0473283i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 2.23iT \)
23 \( 1 + (-0.739 + 22.9i)T \)
good2 \( 1 - 2.24T + 4T^{2} \)
3 \( 1 - 3.98T + 9T^{2} \)
7 \( 1 - 6.68iT - 49T^{2} \)
11 \( 1 + 10.4iT - 121T^{2} \)
13 \( 1 + 4.43T + 169T^{2} \)
17 \( 1 + 1.02iT - 289T^{2} \)
19 \( 1 - 24.8iT - 361T^{2} \)
29 \( 1 - 20.9T + 841T^{2} \)
31 \( 1 - 35.4T + 961T^{2} \)
37 \( 1 - 50.9iT - 1.36e3T^{2} \)
41 \( 1 + 53.5T + 1.68e3T^{2} \)
43 \( 1 + 64.2iT - 1.84e3T^{2} \)
47 \( 1 + 2.03T + 2.20e3T^{2} \)
53 \( 1 - 13.7iT - 2.80e3T^{2} \)
59 \( 1 - 7.09T + 3.48e3T^{2} \)
61 \( 1 + 68.5iT - 3.72e3T^{2} \)
67 \( 1 - 80.9iT - 4.48e3T^{2} \)
71 \( 1 - 117.T + 5.04e3T^{2} \)
73 \( 1 - 38.6T + 5.32e3T^{2} \)
79 \( 1 + 11.9iT - 6.24e3T^{2} \)
83 \( 1 + 10.8iT - 6.88e3T^{2} \)
89 \( 1 - 17.7iT - 7.92e3T^{2} \)
97 \( 1 - 5.84iT - 9.40e3T^{2} \)
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

−13.61958437250944214662180575660, −12.50131438625654704566595983769, −11.81962850504538837788505260560, −9.934927724975666092745066701541, −8.687494183984695755868252792136, −8.320440618831213447636580036578, −6.27427572558153515080439350821, −5.07947672006931817757637609320, −3.64464397223886816945140226772, −2.55697242671464348141796637565, 2.59205441739866088390927249932, 3.74481043190574865520154618564, 4.83367306097087854306880221304, 6.70979924704636568985161061693, 7.71301035821216761669947886538, 9.098402512278363401376714629594, 10.03459666468593901350364673420, 11.50059548082920404976970765516, 12.77977991756213141732097627776, 13.63557137533485443326940959275

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