↑

Properties

Label 2-416-32.21-c1-0-38
Degree 22
Conductor 416416
Sign 0.900+0.434i0.900 + 0.434i
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
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  + (−0.705 + 1.22i)2-s + (3.06 − 1.26i)3-s + (−1.00 − 1.72i)4-s + (0.112 − 0.272i)5-s + (−0.604 + 4.65i)6-s + (−1.80 − 1.80i)7-s + (2.82 − 0.0142i)8-s + (5.66 − 5.66i)9-s + (0.254 + 0.330i)10-s + (−2.10 − 0.872i)11-s + (−5.27 − 4.02i)12-s + (−0.382 − 0.923i)13-s + (3.49 − 0.942i)14-s − 0.978i·15-s + (−1.97 + 3.47i)16-s − 3.28i·17-s + ⋯
L(s)  = 1  + (−0.498 + 0.866i)2-s + (1.76 − 0.733i)3-s + (−0.502 − 0.864i)4-s + (0.0504 − 0.121i)5-s + (−0.246 + 1.89i)6-s + (−0.683 − 0.683i)7-s + (0.999 − 0.00503i)8-s + (1.88 − 1.88i)9-s + (0.0804 + 0.104i)10-s + (−0.635 − 0.263i)11-s + (−1.52 − 1.16i)12-s + (−0.106 − 0.256i)13-s + (0.933 − 0.251i)14-s − 0.252i·15-s + (−0.494 + 0.869i)16-s − 0.796i·17-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.900+0.434i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.900+0.434i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.900+0.434i0.900 + 0.434i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(53,⋅)\chi_{416} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.900+0.434i)(2,\ 416,\ (\ :1/2),\ 0.900 + 0.434i)

Particular Values

L(1)L(1) ≈\approx 1.66480−0.380827i1.66480 - 0.380827i
L(12)L(\frac12) ≈\approx 1.66480−0.380827i1.66480 - 0.380827i
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+(0.705−1.22i)T 1 + (0.705 - 1.22i)T
13 1+(0.382+0.923i)T 1 + (0.382 + 0.923i)T
good3 1+(−3.06+1.26i)T+(2.12−2.12i)T2 1 + (-3.06 + 1.26i)T + (2.12 - 2.12i)T^{2}
5 1+(−0.112+0.272i)T+(−3.53−3.53i)T2 1 + (-0.112 + 0.272i)T + (-3.53 - 3.53i)T^{2}
7 1+(1.80+1.80i)T+7iT2 1 + (1.80 + 1.80i)T + 7iT^{2}
11 1+(2.10+0.872i)T+(7.77+7.77i)T2 1 + (2.10 + 0.872i)T + (7.77 + 7.77i)T^{2}
17 1+3.28iT−17T2 1 + 3.28iT - 17T^{2}
19 1+(−2.88−6.96i)T+(−13.4+13.4i)T2 1 + (-2.88 - 6.96i)T + (-13.4 + 13.4i)T^{2}
23 1+(−1.45+1.45i)T−23iT2 1 + (-1.45 + 1.45i)T - 23iT^{2}
29 1+(7.19−2.98i)T+(20.5−20.5i)T2 1 + (7.19 - 2.98i)T + (20.5 - 20.5i)T^{2}
31 1−1.16T+31T2 1 - 1.16T + 31T^{2}
37 1+(−1.93+4.67i)T+(−26.1−26.1i)T2 1 + (-1.93 + 4.67i)T + (-26.1 - 26.1i)T^{2}
41 1+(1.89−1.89i)T−41iT2 1 + (1.89 - 1.89i)T - 41iT^{2}
43 1+(−9.69−4.01i)T+(30.4+30.4i)T2 1 + (-9.69 - 4.01i)T + (30.4 + 30.4i)T^{2}
47 1−10.8iT−47T2 1 - 10.8iT - 47T^{2}
53 1+(−6.82−2.82i)T+(37.4+37.4i)T2 1 + (-6.82 - 2.82i)T + (37.4 + 37.4i)T^{2}
59 1+(1.41−3.42i)T+(−41.7−41.7i)T2 1 + (1.41 - 3.42i)T + (-41.7 - 41.7i)T^{2}
61 1+(3.10−1.28i)T+(43.1−43.1i)T2 1 + (3.10 - 1.28i)T + (43.1 - 43.1i)T^{2}
67 1+(2.33−0.966i)T+(47.3−47.3i)T2 1 + (2.33 - 0.966i)T + (47.3 - 47.3i)T^{2}
71 1+(−3.12−3.12i)T+71iT2 1 + (-3.12 - 3.12i)T + 71iT^{2}
73 1+(−1.58+1.58i)T−73iT2 1 + (-1.58 + 1.58i)T - 73iT^{2}
79 1−15.9iT−79T2 1 - 15.9iT - 79T^{2}
83 1+(−1.02−2.47i)T+(−58.6+58.6i)T2 1 + (-1.02 - 2.47i)T + (-58.6 + 58.6i)T^{2}
89 1+(5.24+5.24i)T+89iT2 1 + (5.24 + 5.24i)T + 89iT^{2}
97 1+5.34T+97T2 1 + 5.34T + 97T^{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

−10.73257543689573844876353444278, −9.680858746276130612484275115851, −9.254932180065360623497142983438, −8.197664334989477962107673188815, −7.52201728360333047792761597069, −6.98928393763668145356291748406, −5.69038402301926331667775798826, −4.02501046614192924067452124004, −2.87790544769881024505718925353, −1.20483894783509966937687639681, 2.21817099434297888204050207626, 2.88901046458464814499449274698, 3.87867102541299790008120591028, 5.00450393358026652084033995084, 7.09731868709974497543551539918, 8.002790124869597470822274973405, 8.934473687152927753033616971524, 9.332377314654938639370866310934, 10.14362340800335090968849989533, 10.90912200384046963151457890936

Graph of the ZZ-function along the critical line