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Properties

Label 2-968-88.19-c1-0-43
Degree 22
Conductor 968968
Sign 0.958−0.286i0.958 - 0.286i
Analytic cond. 7.729517.72951
Root an. cond. 2.780202.78020
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.831 − 1.14i)2-s + (0.618 + 1.90i)3-s + (−0.618 + 1.90i)4-s + (1.66 − 2.28i)6-s + (2.68 − 0.874i)8-s + (−0.809 + 0.587i)9-s − 3.99·12-s + (−3.23 − 2.35i)16-s + (3.32 − 4.57i)17-s + (1.34 + 0.437i)18-s + (8.06 − 2.62i)19-s + (3.32 + 4.57i)24-s + (1.54 + 4.75i)25-s + (3.23 + 2.35i)27-s + 5.65i·32-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (0.356 + 1.09i)3-s + (−0.309 + 0.951i)4-s + (0.678 − 0.934i)6-s + (0.951 − 0.309i)8-s + (−0.269 + 0.195i)9-s − 1.15·12-s + (−0.809 − 0.587i)16-s + (0.806 − 1.10i)17-s + (0.317 + 0.103i)18-s + (1.85 − 0.601i)19-s + (0.678 + 0.934i)24-s + (0.309 + 0.951i)25-s + (0.622 + 0.452i)27-s + 0.999i·32-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=((0.958−0.286i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=((0.958−0.286i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: 0.958−0.286i0.958 - 0.286i
Analytic conductor: 7.729517.72951
Root analytic conductor: 2.780202.78020
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ968(723,⋅)\chi_{968} (723, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 968, ( :1/2), 0.958−0.286i)(2,\ 968,\ (\ :1/2),\ 0.958 - 0.286i)

Particular Values

L(1)L(1) ≈\approx 1.38023+0.201819i1.38023 + 0.201819i
L(12)L(\frac12) ≈\approx 1.38023+0.201819i1.38023 + 0.201819i
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+(0.831+1.14i)T 1 + (0.831 + 1.14i)T
11 1 1
good3 1+(−0.618−1.90i)T+(−2.42+1.76i)T2 1 + (-0.618 - 1.90i)T + (-2.42 + 1.76i)T^{2}
5 1+(−1.54−4.75i)T2 1 + (-1.54 - 4.75i)T^{2}
7 1+(−5.66−4.11i)T2 1 + (-5.66 - 4.11i)T^{2}
13 1+(4.01−12.3i)T2 1 + (4.01 - 12.3i)T^{2}
17 1+(−3.32+4.57i)T+(−5.25−16.1i)T2 1 + (-3.32 + 4.57i)T + (-5.25 - 16.1i)T^{2}
19 1+(−8.06+2.62i)T+(15.3−11.1i)T2 1 + (-8.06 + 2.62i)T + (15.3 - 11.1i)T^{2}
23 1−23T2 1 - 23T^{2}
29 1+(−23.4−17.0i)T2 1 + (-23.4 - 17.0i)T^{2}
31 1+(−9.57+29.4i)T2 1 + (-9.57 + 29.4i)T^{2}
37 1+(29.9+21.7i)T2 1 + (29.9 + 21.7i)T^{2}
41 1+(10.7−3.49i)T+(33.1−24.0i)T2 1 + (10.7 - 3.49i)T + (33.1 - 24.0i)T^{2}
43 1−8.48iT−43T2 1 - 8.48iT - 43T^{2}
47 1+(38.0−27.6i)T2 1 + (38.0 - 27.6i)T^{2}
53 1+(−16.3+50.4i)T2 1 + (-16.3 + 50.4i)T^{2}
59 1+(1.85−5.70i)T+(−47.7−34.6i)T2 1 + (1.85 - 5.70i)T + (-47.7 - 34.6i)T^{2}
61 1+(18.8+58.0i)T2 1 + (18.8 + 58.0i)T^{2}
67 1−14T+67T2 1 - 14T + 67T^{2}
71 1+(−21.9−67.5i)T2 1 + (-21.9 - 67.5i)T^{2}
73 1+(16.1+5.24i)T+(59.0+42.9i)T2 1 + (16.1 + 5.24i)T + (59.0 + 42.9i)T^{2}
79 1+(24.4−75.1i)T2 1 + (24.4 - 75.1i)T^{2}
83 1+(−1.66+2.28i)T+(−25.6−78.9i)T2 1 + (-1.66 + 2.28i)T + (-25.6 - 78.9i)T^{2}
89 1−18T+89T2 1 - 18T + 89T^{2}
97 1+(8.09−5.87i)T+(29.9−92.2i)T2 1 + (8.09 - 5.87i)T + (29.9 - 92.2i)T^{2}
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   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

−9.812485814054689917506477541590, −9.510776758998169859136161456581, −8.795900215282151910972457928209, −7.71535688597177114058285562389, −7.02642726856748999922570370698, −5.29938131556421681876797322014, −4.60747256774488109939658552174, −3.39137474446855105255960182977, −2.94581417892924263598829772378, −1.15886717318384733764948724713, 0.999877725211212414818740559137, 2.02435868604398995908738758780, 3.60187525345255533036264675113, 5.08060657767811712694530541479, 5.92364350865537645902763071413, 6.83250805199909928764923346610, 7.49789368704775662339885992421, 8.160169577003454342218514473197, 8.793239240651029431098835156546, 9.983306128581646517250115318379

Graph of the ZZ-function along the critical line