↑

Properties

Label 2-3240-9.4-c1-0-4
Degree 22
Conductor 32403240
Sign −0.939−0.342i-0.939 - 0.342i
Analytic cond. 25.871525.8715
Root an. cond. 5.086405.08640
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  + (0.5 − 0.866i)5-s + (2 + 3.46i)7-s + (−1 − 1.73i)11-s + (−2 + 3.46i)13-s + 17-s − 5·19-s + (−2.5 + 4.33i)23-s + (−0.499 − 0.866i)25-s + (−4 − 6.92i)29-s + (−3.5 + 6.06i)31-s + 3.99·35-s − 6·37-s + (−3 + 5.19i)41-s + (1 + 1.73i)43-s + (−4 − 6.92i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.755 + 1.30i)7-s + (−0.301 − 0.522i)11-s + (−0.554 + 0.960i)13-s + 0.242·17-s − 1.14·19-s + (−0.521 + 0.902i)23-s + (−0.0999 − 0.173i)25-s + (−0.742 − 1.28i)29-s + (−0.628 + 1.08i)31-s + 0.676·35-s − 0.986·37-s + (−0.468 + 0.811i)41-s + (0.152 + 0.264i)43-s + (−0.583 − 1.01i)47-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((−0.939−0.342i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s+1/2) L(s)=((−0.939−0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: −0.939−0.342i-0.939 - 0.342i
Analytic conductor: 25.871525.8715
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3240(1081,⋅)\chi_{3240} (1081, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :1/2), −0.939−0.342i)(2,\ 3240,\ (\ :1/2),\ -0.939 - 0.342i)

Particular Values

L(1)L(1) ≈\approx 0.64714004890.6471400489
L(12)L(\frac12) ≈\approx 0.64714004890.6471400489
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)
bad2 1 1
3 1 1
5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good7 1+(−2−3.46i)T+(−3.5+6.06i)T2 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2}
11 1+(1+1.73i)T+(−5.5+9.52i)T2 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2}
13 1+(2−3.46i)T+(−6.5−11.2i)T2 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2}
17 1−T+17T2 1 - T + 17T^{2}
19 1+5T+19T2 1 + 5T + 19T^{2}
23 1+(2.5−4.33i)T+(−11.5−19.9i)T2 1 + (2.5 - 4.33i)T + (-11.5 - 19.9i)T^{2}
29 1+(4+6.92i)T+(−14.5+25.1i)T2 1 + (4 + 6.92i)T + (-14.5 + 25.1i)T^{2}
31 1+(3.5−6.06i)T+(−15.5−26.8i)T2 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1+(3−5.19i)T+(−20.5−35.5i)T2 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2}
43 1+(−1−1.73i)T+(−21.5+37.2i)T2 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2}
47 1+(4+6.92i)T+(−23.5+40.7i)T2 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2}
53 1−9T+53T2 1 - 9T + 53T^{2}
59 1+(2−3.46i)T+(−29.5−51.0i)T2 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2}
61 1+(6.5+11.2i)T+(−30.5+52.8i)T2 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5+8.66i)T+(−33.5−58.0i)T2 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1+(4.5+7.79i)T+(−39.5+68.4i)T2 1 + (4.5 + 7.79i)T + (-39.5 + 68.4i)T^{2}
83 1+(−8.5−14.7i)T+(−41.5+71.8i)T2 1 + (-8.5 - 14.7i)T + (-41.5 + 71.8i)T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+(−4−6.92i)T+(−48.5+84.0i)T2 1 + (-4 - 6.92i)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

−8.904140330653818337165178648858, −8.367117944663208171186397838457, −7.66903149009137062757434750620, −6.63579768194865165219740901249, −5.83065703710561033975219988472, −5.23676594262831427400264063527, −4.51834540101091665565037881762, −3.44867414393349377751825320280, −2.21756807244551354036819310769, −1.73362831597264875195716787381, 0.17931063121907144766927808265, 1.59253611192481790698860195029, 2.53461070912300990198204588456, 3.68260359235524166952283971697, 4.41078073699718774615675132324, 5.20294727933758717150986698979, 6.04419303541784649137779314547, 7.17073448359366060728582154545, 7.36661213231094987205221661183, 8.216014257130467112720472354466

Graph of the ZZ-function along the critical line