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Properties

Label 2-294-21.20-c5-0-12
Degree 22
Conductor 294294
Sign −0.373−0.927i-0.373 - 0.927i
Analytic cond. 47.152847.1528
Root an. cond. 6.866796.86679
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + (−10.8 + 11.2i)3-s − 16·4-s − 104.·5-s + (−44.9 − 43.2i)6-s − 64i·8-s + (−9.33 − 242. i)9-s − 419. i·10-s − 83.8i·11-s + (172. − 179. i)12-s + 382. i·13-s + (1.13e3 − 1.17e3i)15-s + 256·16-s − 1.43e3·17-s + (971. − 37.3i)18-s − 2.24e3i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.693 + 0.720i)3-s − 0.5·4-s − 1.87·5-s + (−0.509 − 0.490i)6-s − 0.353i·8-s + (−0.0384 − 0.999i)9-s − 1.32i·10-s − 0.208i·11-s + (0.346 − 0.360i)12-s + 0.628i·13-s + (1.29 − 1.35i)15-s + 0.250·16-s − 1.20·17-s + (0.706 − 0.0271i)18-s − 1.42i·19-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((−0.373−0.927i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=((−0.373−0.927i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: −0.373−0.927i-0.373 - 0.927i
Analytic conductor: 47.152847.1528
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ294(293,⋅)\chi_{294} (293, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 294, ( :5/2), −0.373−0.927i)(2,\ 294,\ (\ :5/2),\ -0.373 - 0.927i)

Particular Values

L(3)L(3) ≈\approx 0.34392568390.3439256839
L(12)L(\frac12) ≈\approx 0.34392568390.3439256839
L(72)L(\frac{7}{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 1−4iT 1 - 4iT
3 1+(10.8−11.2i)T 1 + (10.8 - 11.2i)T
7 1 1
good5 1+104.T+3.12e3T2 1 + 104.T + 3.12e3T^{2}
11 1+83.8iT−1.61e5T2 1 + 83.8iT - 1.61e5T^{2}
13 1−382.iT−3.71e5T2 1 - 382. iT - 3.71e5T^{2}
17 1+1.43e3T+1.41e6T2 1 + 1.43e3T + 1.41e6T^{2}
19 1+2.24e3iT−2.47e6T2 1 + 2.24e3iT - 2.47e6T^{2}
23 1+4.69e3iT−6.43e6T2 1 + 4.69e3iT - 6.43e6T^{2}
29 1−3.17e3iT−2.05e7T2 1 - 3.17e3iT - 2.05e7T^{2}
31 1−2.36e3iT−2.86e7T2 1 - 2.36e3iT - 2.86e7T^{2}
37 1+1.14e4T+6.93e7T2 1 + 1.14e4T + 6.93e7T^{2}
41 1+3.03e3T+1.15e8T2 1 + 3.03e3T + 1.15e8T^{2}
43 1+1.28e4T+1.47e8T2 1 + 1.28e4T + 1.47e8T^{2}
47 1+4.80e3T+2.29e8T2 1 + 4.80e3T + 2.29e8T^{2}
53 1−858.iT−4.18e8T2 1 - 858. iT - 4.18e8T^{2}
59 1+3.19e4T+7.14e8T2 1 + 3.19e4T + 7.14e8T^{2}
61 1−2.90e4iT−8.44e8T2 1 - 2.90e4iT - 8.44e8T^{2}
67 1−3.70e4T+1.35e9T2 1 - 3.70e4T + 1.35e9T^{2}
71 1+3.39e4iT−1.80e9T2 1 + 3.39e4iT - 1.80e9T^{2}
73 1+5.24e4iT−2.07e9T2 1 + 5.24e4iT - 2.07e9T^{2}
79 1+4.06e4T+3.07e9T2 1 + 4.06e4T + 3.07e9T^{2}
83 1+1.58e4T+3.93e9T2 1 + 1.58e4T + 3.93e9T^{2}
89 1+1.44e5T+5.58e9T2 1 + 1.44e5T + 5.58e9T^{2}
97 1+7.65e4iT−8.58e9T2 1 + 7.65e4iT - 8.58e9T^{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

−11.20545867458418565883333107380, −10.53367108472170535745678608379, −8.924620524802212093858793500480, −8.548228097463028054744212014163, −7.06287602753348041771750186843, −6.59678603115269440921519170358, −4.82545440493059264371289882214, −4.44571681934408645338207778002, −3.28127376128120027420217036433, −0.44720427278187012541325096198, 0.25856805584792329759853516258, 1.64345601480952341771490549808, 3.31738712515443015350632592539, 4.30359678062316363772355705700, 5.46249849096673015204109100532, 6.92355319503798119145526761319, 7.82443331517056686957039510968, 8.434018673047925058057673732121, 10.00628463283514185154544758066, 11.10624525611457227411542643411

Graph of the ZZ-function along the critical line