↑

Properties

Label 2-352-44.19-c1-0-9
Degree 22
Conductor 352352
Sign 0.906+0.421i0.906 + 0.421i
Analytic cond. 2.810732.81073
Root an. cond. 1.676521.67652
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  + (2.26 − 0.734i)3-s + (0.0210 + 0.0153i)5-s + (0.140 − 0.432i)7-s + (2.14 − 1.55i)9-s + (3.11 + 1.12i)11-s + (−1.27 − 1.76i)13-s + (0.0589 + 0.0191i)15-s + (1.94 − 2.67i)17-s + (0.802 + 2.47i)19-s − 1.07i·21-s + 3.69i·23-s + (−1.54 − 4.75i)25-s + (−0.488 + 0.671i)27-s + (−9.31 − 3.02i)29-s + (1.81 + 2.49i)31-s + ⋯
L(s)  = 1  + (1.30 − 0.424i)3-s + (0.00943 + 0.00685i)5-s + (0.0530 − 0.163i)7-s + (0.714 − 0.519i)9-s + (0.940 + 0.340i)11-s + (−0.354 − 0.488i)13-s + (0.0152 + 0.00494i)15-s + (0.470 − 0.647i)17-s + (0.184 + 0.566i)19-s − 0.235i·21-s + 0.770i·23-s + (−0.308 − 0.950i)25-s + (−0.0939 + 0.129i)27-s + (−1.73 − 0.562i)29-s + (0.325 + 0.448i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.906+0.421i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.906+0.421i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.906+0.421i0.906 + 0.421i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(63,⋅)\chi_{352} (63, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.906+0.421i)(2,\ 352,\ (\ :1/2),\ 0.906 + 0.421i)

Particular Values

L(1)L(1) ≈\approx 1.97335−0.435873i1.97335 - 0.435873i
L(12)L(\frac12) ≈\approx 1.97335−0.435873i1.97335 - 0.435873i
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
11 1+(−3.11−1.12i)T 1 + (-3.11 - 1.12i)T
good3 1+(−2.26+0.734i)T+(2.42−1.76i)T2 1 + (-2.26 + 0.734i)T + (2.42 - 1.76i)T^{2}
5 1+(−0.0210−0.0153i)T+(1.54+4.75i)T2 1 + (-0.0210 - 0.0153i)T + (1.54 + 4.75i)T^{2}
7 1+(−0.140+0.432i)T+(−5.66−4.11i)T2 1 + (-0.140 + 0.432i)T + (-5.66 - 4.11i)T^{2}
13 1+(1.27+1.76i)T+(−4.01+12.3i)T2 1 + (1.27 + 1.76i)T + (-4.01 + 12.3i)T^{2}
17 1+(−1.94+2.67i)T+(−5.25−16.1i)T2 1 + (-1.94 + 2.67i)T + (-5.25 - 16.1i)T^{2}
19 1+(−0.802−2.47i)T+(−15.3+11.1i)T2 1 + (-0.802 - 2.47i)T + (-15.3 + 11.1i)T^{2}
23 1−3.69iT−23T2 1 - 3.69iT - 23T^{2}
29 1+(9.31+3.02i)T+(23.4+17.0i)T2 1 + (9.31 + 3.02i)T + (23.4 + 17.0i)T^{2}
31 1+(−1.81−2.49i)T+(−9.57+29.4i)T2 1 + (-1.81 - 2.49i)T + (-9.57 + 29.4i)T^{2}
37 1+(2.27−7.00i)T+(−29.9−21.7i)T2 1 + (2.27 - 7.00i)T + (-29.9 - 21.7i)T^{2}
41 1+(3.59−1.16i)T+(33.1−24.0i)T2 1 + (3.59 - 1.16i)T + (33.1 - 24.0i)T^{2}
43 1+10.2T+43T2 1 + 10.2T + 43T^{2}
47 1+(−10.7+3.47i)T+(38.0−27.6i)T2 1 + (-10.7 + 3.47i)T + (38.0 - 27.6i)T^{2}
53 1+(−7.06+5.13i)T+(16.3−50.4i)T2 1 + (-7.06 + 5.13i)T + (16.3 - 50.4i)T^{2}
59 1+(6.16+2.00i)T+(47.7+34.6i)T2 1 + (6.16 + 2.00i)T + (47.7 + 34.6i)T^{2}
61 1+(7.78−10.7i)T+(−18.8−58.0i)T2 1 + (7.78 - 10.7i)T + (-18.8 - 58.0i)T^{2}
67 1−8.31iT−67T2 1 - 8.31iT - 67T^{2}
71 1+(−2.37+3.26i)T+(−21.9−67.5i)T2 1 + (-2.37 + 3.26i)T + (-21.9 - 67.5i)T^{2}
73 1+(3.48+1.13i)T+(59.0+42.9i)T2 1 + (3.48 + 1.13i)T + (59.0 + 42.9i)T^{2}
79 1+(−2.29+1.66i)T+(24.4−75.1i)T2 1 + (-2.29 + 1.66i)T + (24.4 - 75.1i)T^{2}
83 1+(−1.28−0.934i)T+(25.6+78.9i)T2 1 + (-1.28 - 0.934i)T + (25.6 + 78.9i)T^{2}
89 1+11.9T+89T2 1 + 11.9T + 89T^{2}
97 1+(−0.120+0.0875i)T+(29.9−92.2i)T2 1 + (-0.120 + 0.0875i)T + (29.9 - 92.2i)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

−11.65794466257011388395348569629, −10.19108165941962337241855261041, −9.481712322285158818063682031139, −8.559862231524703747543427585816, −7.69156684892840039003614201630, −6.97514734082246030322455857381, −5.55788881836140918897842394385, −4.04090732274860988021536861619, −3.00460832261600005842325277960, −1.66230075470952485835856993064, 1.96209150963325959685292842980, 3.34503207207568131748700701441, 4.16267140549810298121270510965, 5.62538884593340010985027652882, 6.96396957888349761723800930162, 7.954954574357537728764657013894, 9.069023838473556106727934820584, 9.251884669551878705970602722293, 10.48119967795074845648770327790, 11.51989012916702720551200537656

Graph of the ZZ-function along the critical line