↑

Properties

Label 2-45-45.13-c2-0-8
Degree $2$
Conductor $45$
Sign $-0.971 + 0.238i$
Analytic cond. $1.22616$
Root an. cond. $1.10732$
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  + (−3.22 − 0.863i)2-s + (1.08 − 2.79i)3-s + (6.18 + 3.57i)4-s + (−4.88 − 1.04i)5-s + (−5.91 + 8.07i)6-s + (−8.79 − 2.35i)7-s + (−7.41 − 7.41i)8-s + (−6.64 − 6.07i)9-s + (14.8 + 7.58i)10-s + (1.02 + 1.77i)11-s + (16.7 − 13.4i)12-s + (4.18 − 1.12i)13-s + (26.3 + 15.2i)14-s + (−8.22 + 12.5i)15-s + (3.22 + 5.57i)16-s + (17.1 − 17.1i)17-s + ⋯
L(s)  = 1  + (−1.61 − 0.431i)2-s + (0.362 − 0.932i)3-s + (1.54 + 0.892i)4-s + (−0.977 − 0.208i)5-s + (−0.986 + 1.34i)6-s + (−1.25 − 0.336i)7-s + (−0.927 − 0.927i)8-s + (−0.737 − 0.675i)9-s + (1.48 + 0.758i)10-s + (0.0930 + 0.161i)11-s + (1.39 − 1.11i)12-s + (0.321 − 0.0862i)13-s + (1.88 + 1.08i)14-s + (−0.548 + 0.836i)15-s + (0.201 + 0.348i)16-s + (1.00 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.971 + 0.238i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ -0.971 + 0.238i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0406383 - 0.336143i\)
\(L(\frac12)\) \(\approx\) \(0.0406383 - 0.336143i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.08 + 2.79i)T \)
5 \( 1 + (4.88 + 1.04i)T \)
good2 \( 1 + (3.22 + 0.863i)T + (3.46 + 2i)T^{2} \)
7 \( 1 + (8.79 + 2.35i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-1.02 - 1.77i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.18 + 1.12i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (-17.1 + 17.1i)T - 289iT^{2} \)
19 \( 1 + 18.6iT - 361T^{2} \)
23 \( 1 + (11.5 - 3.08i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (-19.4 + 11.2i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-11.9 + 20.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (33.2 - 33.2i)T - 1.36e3iT^{2} \)
41 \( 1 + (-2.15 + 3.73i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-8.15 + 30.4i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (0.838 + 0.224i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (3.33 + 3.33i)T + 2.80e3iT^{2} \)
59 \( 1 + (66.3 + 38.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-35.3 - 61.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-5.59 - 20.8i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 28.6T + 5.04e3T^{2} \)
73 \( 1 + (-48.1 - 48.1i)T + 5.32e3iT^{2} \)
79 \( 1 + (0.201 - 0.116i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-41.1 + 153. i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 103. iT - 7.92e3T^{2} \)
97 \( 1 + (-69.7 - 18.6i)T + (8.14e3 + 4.70e3i)T^{2} \)
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

−15.57268649895245427781330644693, −13.67878679620389046984070015632, −12.34089183864497649773623653368, −11.51030747660983714678207045266, −9.955073255451270083893470361007, −8.855589758395311567468516508338, −7.71612950834384212933265118849, −6.80869404128181022893827691997, −3.08070734412108695996445900064, −0.57053455152977856133521748765, 3.51931557248201998018766181665, 6.23906578099585294868140547233, 7.87314263355238579058609342242, 8.789218436905404506358130525330, 9.954188156580867648381682009082, 10.73748782817189492183858814685, 12.30089303369130487846259545695, 14.44693726568797066594118551670, 15.63548596176604047588553836921, 16.13035593503575297361664060069

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