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Properties

Label 2-352-88.43-c1-0-1
Degree 22
Conductor 352352
Sign 0.426−0.904i0.426 - 0.904i
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s + (3 + 1.41i)11-s + 5.65i·17-s + 8.48i·19-s + 5·25-s + 4·27-s + (−6 − 2.82i)33-s + 11.3i·41-s − 8.48i·43-s − 7·49-s − 11.3i·51-s − 16.9i·57-s + 6·59-s − 14·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.333·9-s + (0.904 + 0.426i)11-s + 1.37i·17-s + 1.94i·19-s + 25-s + 0.769·27-s + (−1.04 − 0.492i)33-s + 1.76i·41-s − 1.29i·43-s − 49-s − 1.58i·51-s − 2.24i·57-s + 0.781·59-s − 1.71·67-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.426−0.904i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.426−0.904i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.426−0.904i0.426 - 0.904i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(175,⋅)\chi_{352} (175, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.426−0.904i)(2,\ 352,\ (\ :1/2),\ 0.426 - 0.904i)

Particular Values

L(1)L(1) ≈\approx 0.694545+0.440437i0.694545 + 0.440437i
L(12)L(\frac12) ≈\approx 0.694545+0.440437i0.694545 + 0.440437i
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+(−3−1.41i)T 1 + (-3 - 1.41i)T
good3 1+2T+3T2 1 + 2T + 3T^{2}
5 1−5T2 1 - 5T^{2}
7 1+7T2 1 + 7T^{2}
13 1+13T2 1 + 13T^{2}
17 1−5.65iT−17T2 1 - 5.65iT - 17T^{2}
19 1−8.48iT−19T2 1 - 8.48iT - 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1−11.3iT−41T2 1 - 11.3iT - 41T^{2}
43 1+8.48iT−43T2 1 + 8.48iT - 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1−6T+59T2 1 - 6T + 59T^{2}
61 1+61T2 1 + 61T^{2}
67 1+14T+67T2 1 + 14T + 67T^{2}
71 1−71T2 1 - 71T^{2}
73 1+16.9iT−73T2 1 + 16.9iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+2.82iT−83T2 1 + 2.82iT - 83T^{2}
89 1−18T+89T2 1 - 18T + 89T^{2}
97 1−10T+97T2 1 - 10T + 97T^{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

−11.78564837200667383413496578252, −10.69630340124542091396161148176, −10.12212387802391454264212625168, −8.878008511181446347730900255814, −7.83628813223193838458185988316, −6.50066287610632994931038182472, −5.98742920335379963792988417187, −4.79528488318644548625366106906, −3.62988165438279953775200850743, −1.51828845451592432983997148942, 0.71747212566875794729493289639, 2.90341150757303136225476350856, 4.56185882949164792426677026649, 5.37579875239465540589284238388, 6.52660976618871657974873541775, 7.14693182699885038244888899715, 8.715914008706220274171438711728, 9.437006067333637949281575212086, 10.71754742669555316812319664663, 11.38335954939917094632035276231

Graph of the ZZ-function along the critical line