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Properties

Label 2-451-451.119-c1-0-3
Degree 22
Conductor 451451
Sign 0.984+0.175i0.984 + 0.175i
Analytic cond. 3.601253.60125
Root an. cond. 1.897691.89769
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  + (−0.804 − 2.47i)2-s + (0.229 − 0.166i)3-s + (−3.86 + 2.80i)4-s − 1.16·5-s + (−0.596 − 0.433i)6-s + (−0.525 − 1.61i)7-s + (5.85 + 4.25i)8-s + (−0.902 + 2.77i)9-s + (0.941 + 2.89i)10-s + (2.26 + 2.42i)11-s + (−0.418 + 1.28i)12-s + (1.00 + 3.08i)13-s + (−3.58 + 2.60i)14-s + (−0.267 + 0.194i)15-s + (2.86 − 8.81i)16-s + (−6.43 + 4.67i)17-s + ⋯
L(s)  = 1  + (−0.568 − 1.75i)2-s + (0.132 − 0.0960i)3-s + (−1.93 + 1.40i)4-s − 0.523·5-s + (−0.243 − 0.176i)6-s + (−0.198 − 0.611i)7-s + (2.06 + 1.50i)8-s + (−0.300 + 0.925i)9-s + (0.297 + 0.915i)10-s + (0.683 + 0.730i)11-s + (−0.120 + 0.371i)12-s + (0.278 + 0.856i)13-s + (−0.957 + 0.695i)14-s + (−0.0691 + 0.0502i)15-s + (0.716 − 2.20i)16-s + (−1.56 + 1.13i)17-s + ⋯

Functional equation

Λ(s)=(451s/2 ΓC(s) L(s)=((0.984+0.175i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(451s/2 ΓC(s+1/2) L(s)=((0.984+0.175i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 451451    =    11⋅4111 \cdot 41
Sign: 0.984+0.175i0.984 + 0.175i
Analytic conductor: 3.601253.60125
Root analytic conductor: 1.897691.89769
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ451(119,⋅)\chi_{451} (119, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 451, ( :1/2), 0.984+0.175i)(2,\ 451,\ (\ :1/2),\ 0.984 + 0.175i)

Particular Values

L(1)L(1) ≈\approx 0.572696−0.0505721i0.572696 - 0.0505721i
L(12)L(\frac12) ≈\approx 0.572696−0.0505721i0.572696 - 0.0505721i
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)
bad11 1+(−2.26−2.42i)T 1 + (-2.26 - 2.42i)T
41 1+(0.000300−6.40i)T 1 + (0.000300 - 6.40i)T
good2 1+(0.804+2.47i)T+(−1.61+1.17i)T2 1 + (0.804 + 2.47i)T + (-1.61 + 1.17i)T^{2}
3 1+(−0.229+0.166i)T+(0.927−2.85i)T2 1 + (-0.229 + 0.166i)T + (0.927 - 2.85i)T^{2}
5 1+1.16T+5T2 1 + 1.16T + 5T^{2}
7 1+(0.525+1.61i)T+(−5.66+4.11i)T2 1 + (0.525 + 1.61i)T + (-5.66 + 4.11i)T^{2}
13 1+(−1.00−3.08i)T+(−10.5+7.64i)T2 1 + (-1.00 - 3.08i)T + (-10.5 + 7.64i)T^{2}
17 1+(6.43−4.67i)T+(5.25−16.1i)T2 1 + (6.43 - 4.67i)T + (5.25 - 16.1i)T^{2}
19 1+0.642T+19T2 1 + 0.642T + 19T^{2}
23 1+(−7.42+5.39i)T+(7.10−21.8i)T2 1 + (-7.42 + 5.39i)T + (7.10 - 21.8i)T^{2}
29 1+(5.26−3.82i)T+(8.96−27.5i)T2 1 + (5.26 - 3.82i)T + (8.96 - 27.5i)T^{2}
31 1−4.54T+31T2 1 - 4.54T + 31T^{2}
37 1+(−7.97+5.79i)T+(11.4−35.1i)T2 1 + (-7.97 + 5.79i)T + (11.4 - 35.1i)T^{2}
43 1+(0.459−0.334i)T+(13.2−40.8i)T2 1 + (0.459 - 0.334i)T + (13.2 - 40.8i)T^{2}
47 1+(3.41−10.4i)T+(−38.0−27.6i)T2 1 + (3.41 - 10.4i)T + (-38.0 - 27.6i)T^{2}
53 1+(5.91+4.29i)T+(16.3+50.4i)T2 1 + (5.91 + 4.29i)T + (16.3 + 50.4i)T^{2}
59 1+7.36T+59T2 1 + 7.36T + 59T^{2}
61 1+(3.18−9.79i)T+(−49.3−35.8i)T2 1 + (3.18 - 9.79i)T + (-49.3 - 35.8i)T^{2}
67 1+(3.14−9.68i)T+(−54.2−39.3i)T2 1 + (3.14 - 9.68i)T + (-54.2 - 39.3i)T^{2}
71 1+(5.12−3.72i)T+(21.9−67.5i)T2 1 + (5.12 - 3.72i)T + (21.9 - 67.5i)T^{2}
73 1+(−7.38−5.36i)T+(22.5+69.4i)T2 1 + (-7.38 - 5.36i)T + (22.5 + 69.4i)T^{2}
79 1+(1.33−4.10i)T+(−63.9−46.4i)T2 1 + (1.33 - 4.10i)T + (-63.9 - 46.4i)T^{2}
83 1+(−2.18−6.73i)T+(−67.1+48.7i)T2 1 + (-2.18 - 6.73i)T + (-67.1 + 48.7i)T^{2}
89 1+(7.43+5.40i)T+(27.5+84.6i)T2 1 + (7.43 + 5.40i)T + (27.5 + 84.6i)T^{2}
97 1+(−12.1−8.85i)T+(29.9+92.2i)T2 1 + (-12.1 - 8.85i)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

−11.10937054385435889714200817005, −10.44674226948582585642404968034, −9.338049125776097903579002572802, −8.722258946903771872025582516668, −7.78204824782912598714782237818, −6.64126431765513692408287835973, −4.47781602661990780621196313215, −4.10981967927298364359376740603, −2.64833933969607813221730672060, −1.53359827781495472349603505729, 0.45322365550991947372457673788, 3.33103542953429442528319784598, 4.68792977433084188435284996972, 5.84115666347562426193124385563, 6.46315037171752346071658483294, 7.43667367740876214943877822056, 8.364886277114328134981884410570, 9.165791779793964740976093203340, 9.470528476306157856232335381871, 11.06846925119390634984008462248

Graph of the ZZ-function along the critical line