↑

Properties

Label 2-45-9.5-c4-0-13
Degree $2$
Conductor $45$
Sign $0.202 + 0.979i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.78 − 2.76i)2-s + (4.38 − 7.86i)3-s + (7.27 − 12.6i)4-s + (9.68 + 5.59i)5-s + (−0.740 − 49.7i)6-s + (−0.820 − 1.42i)7-s + 7.97i·8-s + (−42.5 − 68.9i)9-s + 61.8·10-s + (−46.1 + 26.6i)11-s + (−67.1 − 112. i)12-s + (−30.4 + 52.8i)13-s + (−7.85 − 4.53i)14-s + (86.3 − 51.6i)15-s + (138. + 239. i)16-s + 106. i·17-s + ⋯
L(s)  = 1  + (1.19 − 0.690i)2-s + (0.487 − 0.873i)3-s + (0.454 − 0.787i)4-s + (0.387 + 0.223i)5-s + (−0.0205 − 1.38i)6-s + (−0.0167 − 0.0289i)7-s + 0.124i·8-s + (−0.525 − 0.850i)9-s + 0.618·10-s + (−0.381 + 0.220i)11-s + (−0.466 − 0.781i)12-s + (−0.180 + 0.312i)13-s + (−0.0400 − 0.0231i)14-s + (0.383 − 0.229i)15-s + (0.541 + 0.937i)16-s + 0.369i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.202 + 0.979i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.202 + 0.979i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.36585 - 1.92595i\)
\(L(\frac12)\) \(\approx\) \(2.36585 - 1.92595i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.38 + 7.86i)T \)
5 \( 1 + (-9.68 - 5.59i)T \)
good2 \( 1 + (-4.78 + 2.76i)T + (8 - 13.8i)T^{2} \)
7 \( 1 + (0.820 + 1.42i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (46.1 - 26.6i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (30.4 - 52.8i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 106. iT - 8.35e4T^{2} \)
19 \( 1 - 396.T + 1.30e5T^{2} \)
23 \( 1 + (213. + 123. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-9.99e2 + 577. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (761. - 1.31e3i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 2.71e3T + 1.87e6T^{2} \)
41 \( 1 + (2.00e3 + 1.15e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.46 + 2.54i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.65e3 + 1.53e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 4.00e3iT - 7.89e6T^{2} \)
59 \( 1 + (5.83e3 + 3.36e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-1.19e3 - 2.07e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (586. - 1.01e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 4.26e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.72e3T + 2.83e7T^{2} \)
79 \( 1 + (-3.76e3 - 6.51e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (2.93e3 - 1.69e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 2.31e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.57e3 - 2.73e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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

−14.15908828808825827531512611716, −13.75171513592492462339479279898, −12.55673325859534770571615246328, −11.82411296968373041738262603300, −10.29927711128853379297671892792, −8.532339739886391514811261480645, −6.91227304704715947985941039565, −5.35217479488735510813172982180, −3.40152459714341816511161675581, −1.97369172166833137044038059749, 3.18572167124461586606293339315, 4.75296031374576677011689928150, 5.75607852931321986293150490363, 7.59000306495350072481999990505, 9.228691530052115074786421632073, 10.43021078593199042189761298171, 12.16843618536851079016117918782, 13.58589609438734362878058327776, 14.09007600925418035016256052857, 15.33148936830682318737852904718

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