↑

Properties

Label 2-45-45.34-c1-0-3
Degree 22
Conductor 4545
Sign 0.653+0.757i0.653 + 0.757i
Analytic cond. 0.3593260.359326
Root an. cond. 0.5994380.599438
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.448 − 0.258i)2-s + (0.448 − 1.67i)3-s + (−0.866 − 1.5i)4-s + (−0.358 + 2.20i)5-s + (−0.633 + 0.633i)6-s + (2.89 + 1.67i)7-s + 1.93i·8-s + (−2.59 − 1.50i)9-s + (0.732 − 0.896i)10-s + (−0.633 + 1.09i)11-s + (−2.89 + 0.776i)12-s + (−2.12 + 1.22i)13-s + (−0.866 − 1.5i)14-s + (3.53 + 1.58i)15-s + (−1.23 + 2.13i)16-s − 5.27i·17-s + ⋯
L(s)  = 1  + (−0.316 − 0.183i)2-s + (0.258 − 0.965i)3-s + (−0.433 − 0.750i)4-s + (−0.160 + 0.987i)5-s + (−0.258 + 0.258i)6-s + (1.09 + 0.632i)7-s + 0.683i·8-s + (−0.866 − 0.5i)9-s + (0.231 − 0.283i)10-s + (−0.191 + 0.331i)11-s + (−0.836 + 0.224i)12-s + (−0.588 + 0.339i)13-s + (−0.231 − 0.400i)14-s + (0.911 + 0.410i)15-s + (−0.308 + 0.533i)16-s − 1.28i·17-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((0.653+0.757i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+1/2) L(s)=((0.653+0.757i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: 0.653+0.757i0.653 + 0.757i
Analytic conductor: 0.3593260.359326
Root analytic conductor: 0.5994380.599438
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ45(34,⋅)\chi_{45} (34, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 45, ( :1/2), 0.653+0.757i)(2,\ 45,\ (\ :1/2),\ 0.653 + 0.757i)

Particular Values

L(1)L(1) ≈\approx 0.640006−0.293230i0.640006 - 0.293230i
L(12)L(\frac12) ≈\approx 0.640006−0.293230i0.640006 - 0.293230i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−0.448+1.67i)T 1 + (-0.448 + 1.67i)T
5 1+(0.358−2.20i)T 1 + (0.358 - 2.20i)T
good2 1+(0.448+0.258i)T+(1+1.73i)T2 1 + (0.448 + 0.258i)T + (1 + 1.73i)T^{2}
7 1+(−2.89−1.67i)T+(3.5+6.06i)T2 1 + (-2.89 - 1.67i)T + (3.5 + 6.06i)T^{2}
11 1+(0.633−1.09i)T+(−5.5−9.52i)T2 1 + (0.633 - 1.09i)T + (-5.5 - 9.52i)T^{2}
13 1+(2.12−1.22i)T+(6.5−11.2i)T2 1 + (2.12 - 1.22i)T + (6.5 - 11.2i)T^{2}
17 1+5.27iT−17T2 1 + 5.27iT - 17T^{2}
19 1−0.732T+19T2 1 - 0.732T + 19T^{2}
23 1+(−0.448+0.258i)T+(11.5−19.9i)T2 1 + (-0.448 + 0.258i)T + (11.5 - 19.9i)T^{2}
29 1+(0.232−0.401i)T+(−14.5−25.1i)T2 1 + (0.232 - 0.401i)T + (-14.5 - 25.1i)T^{2}
31 1+(0.366+0.633i)T+(−15.5+26.8i)T2 1 + (0.366 + 0.633i)T + (-15.5 + 26.8i)T^{2}
37 1−4.24iT−37T2 1 - 4.24iT - 37T^{2}
41 1+(3.86+6.69i)T+(−20.5+35.5i)T2 1 + (3.86 + 6.69i)T + (-20.5 + 35.5i)T^{2}
43 1+(−0.568−0.328i)T+(21.5+37.2i)T2 1 + (-0.568 - 0.328i)T + (21.5 + 37.2i)T^{2}
47 1+(2.56+1.48i)T+(23.5+40.7i)T2 1 + (2.56 + 1.48i)T + (23.5 + 40.7i)T^{2}
53 1+1.03iT−53T2 1 + 1.03iT - 53T^{2}
59 1+(−4.73−8.19i)T+(−29.5+51.0i)T2 1 + (-4.73 - 8.19i)T + (-29.5 + 51.0i)T^{2}
61 1+(−3.33+5.76i)T+(−30.5−52.8i)T2 1 + (-3.33 + 5.76i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.57+3.79i)T+(33.5−58.0i)T2 1 + (-6.57 + 3.79i)T + (33.5 - 58.0i)T^{2}
71 1−14.1T+71T2 1 - 14.1T + 71T^{2}
73 1−8.48iT−73T2 1 - 8.48iT - 73T^{2}
79 1+(−3.73+6.46i)T+(−39.5−68.4i)T2 1 + (-3.73 + 6.46i)T + (-39.5 - 68.4i)T^{2}
83 1+(−6.90−3.98i)T+(41.5+71.8i)T2 1 + (-6.90 - 3.98i)T + (41.5 + 71.8i)T^{2}
89 1+13.3T+89T2 1 + 13.3T + 89T^{2}
97 1+(13.1+7.58i)T+(48.5+84.0i)T2 1 + (13.1 + 7.58i)T + (48.5 + 84.0i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−15.25765265406797709493756476543, −14.47184066230641958536629819428, −13.75302340938096798534128091001, −11.96952086924739140056256899157, −11.13133883190253041768959290440, −9.601069534051500250745263584444, −8.242180045575752546111058148941, −6.95480278415243523552271718165, −5.26400770260162682424791785396, −2.26244947342011222834222574810, 3.93576124535336025159435104885, 5.05925354367494962075770773831, 7.88515502509424774423259794821, 8.504047622228970716637441742687, 9.799972013775995612597336561948, 11.19086060461297595318524635221, 12.61318321986611336932119036402, 13.81243423877750383396943925787, 15.04520594880966610974105421384, 16.28917846486032054758719585802

Graph of the ZZ-function along the critical line