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Properties

Label 2-294-147.104-c1-0-0
Degree 22
Conductor 294294
Sign −0.867+0.497i-0.867 + 0.497i
Analytic cond. 2.347602.34760
Root an. cond. 1.532181.53218
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  + (0.781 + 0.623i)2-s + (−1.32 + 1.11i)3-s + (0.222 + 0.974i)4-s + (−3.37 − 1.62i)5-s + (−1.73 + 0.0435i)6-s + (−0.0659 + 2.64i)7-s + (−0.433 + 0.900i)8-s + (0.519 − 2.95i)9-s + (−1.62 − 3.37i)10-s + (−1.00 − 0.803i)11-s + (−1.38 − 1.04i)12-s + (−5.22 − 4.16i)13-s + (−1.70 + 2.02i)14-s + (6.28 − 1.60i)15-s + (−0.900 + 0.433i)16-s + (−0.308 + 1.35i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (−0.765 + 0.642i)3-s + (0.111 + 0.487i)4-s + (−1.50 − 0.726i)5-s + (−0.706 + 0.0177i)6-s + (−0.0249 + 0.999i)7-s + (−0.153 + 0.318i)8-s + (0.173 − 0.984i)9-s + (−0.514 − 1.06i)10-s + (−0.303 − 0.242i)11-s + (−0.398 − 0.301i)12-s + (−1.44 − 1.15i)13-s + (−0.454 + 0.541i)14-s + (1.62 − 0.413i)15-s + (−0.225 + 0.108i)16-s + (−0.0747 + 0.327i)17-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((−0.867+0.497i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+1/2) L(s)=((−0.867+0.497i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: −0.867+0.497i-0.867 + 0.497i
Analytic conductor: 2.347602.34760
Root analytic conductor: 1.532181.53218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ294(251,⋅)\chi_{294} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 294, ( :1/2), −0.867+0.497i)(2,\ 294,\ (\ :1/2),\ -0.867 + 0.497i)

Particular Values

L(1)L(1) ≈\approx 0.0644852−0.242304i0.0644852 - 0.242304i
L(12)L(\frac12) ≈\approx 0.0644852−0.242304i0.0644852 - 0.242304i
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+(−0.781−0.623i)T 1 + (-0.781 - 0.623i)T
3 1+(1.32−1.11i)T 1 + (1.32 - 1.11i)T
7 1+(0.0659−2.64i)T 1 + (0.0659 - 2.64i)T
good5 1+(3.37+1.62i)T+(3.11+3.90i)T2 1 + (3.37 + 1.62i)T + (3.11 + 3.90i)T^{2}
11 1+(1.00+0.803i)T+(2.44+10.7i)T2 1 + (1.00 + 0.803i)T + (2.44 + 10.7i)T^{2}
13 1+(5.22+4.16i)T+(2.89+12.6i)T2 1 + (5.22 + 4.16i)T + (2.89 + 12.6i)T^{2}
17 1+(0.308−1.35i)T+(−15.3−7.37i)T2 1 + (0.308 - 1.35i)T + (-15.3 - 7.37i)T^{2}
19 1−3.51iT−19T2 1 - 3.51iT - 19T^{2}
23 1+(6.90−1.57i)T+(20.7−9.97i)T2 1 + (6.90 - 1.57i)T + (20.7 - 9.97i)T^{2}
29 1+(−1.86−0.426i)T+(26.1+12.5i)T2 1 + (-1.86 - 0.426i)T + (26.1 + 12.5i)T^{2}
31 1−2.07iT−31T2 1 - 2.07iT - 31T^{2}
37 1+(1.39−6.10i)T+(−33.3−16.0i)T2 1 + (1.39 - 6.10i)T + (-33.3 - 16.0i)T^{2}
41 1+(−4.71−2.27i)T+(25.5+32.0i)T2 1 + (-4.71 - 2.27i)T + (25.5 + 32.0i)T^{2}
43 1+(−9.86+4.75i)T+(26.8−33.6i)T2 1 + (-9.86 + 4.75i)T + (26.8 - 33.6i)T^{2}
47 1+(1.70−2.14i)T+(−10.4−45.8i)T2 1 + (1.70 - 2.14i)T + (-10.4 - 45.8i)T^{2}
53 1+(14.1−3.22i)T+(47.7−22.9i)T2 1 + (14.1 - 3.22i)T + (47.7 - 22.9i)T^{2}
59 1+(2.50−1.20i)T+(36.7−46.1i)T2 1 + (2.50 - 1.20i)T + (36.7 - 46.1i)T^{2}
61 1+(0.509+0.116i)T+(54.9+26.4i)T2 1 + (0.509 + 0.116i)T + (54.9 + 26.4i)T^{2}
67 1+9.25T+67T2 1 + 9.25T + 67T^{2}
71 1+(−9.61+2.19i)T+(63.9−30.8i)T2 1 + (-9.61 + 2.19i)T + (63.9 - 30.8i)T^{2}
73 1+(−6.25+4.98i)T+(16.2−71.1i)T2 1 + (-6.25 + 4.98i)T + (16.2 - 71.1i)T^{2}
79 1+0.520T+79T2 1 + 0.520T + 79T^{2}
83 1+(5.43+6.81i)T+(−18.4+80.9i)T2 1 + (5.43 + 6.81i)T + (-18.4 + 80.9i)T^{2}
89 1+(−3.62−4.54i)T+(−19.8+86.7i)T2 1 + (-3.62 - 4.54i)T + (-19.8 + 86.7i)T^{2}
97 1+0.909iT−97T2 1 + 0.909iT - 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

−12.32872490878190259990959644028, −11.76348947135627888965388988948, −10.63839534560078641029538430403, −9.466691920807546235783454453641, −8.242247106831826835028931545615, −7.65197538916245610027636143668, −6.07122299908982143087002355127, −5.19821189164580202073222282369, −4.39286975048049044249817795873, −3.20058636099949165475265285335, 0.16239757101005030999019189674, 2.44053255303087753124971815005, 4.09486743687396872401427958826, 4.74884904795076866715528489803, 6.47799723170091932137934116916, 7.26889107756393991627336397784, 7.78467607714389781350966208910, 9.720462764991687754284179491828, 10.79153416512118445363510584744, 11.32518105873153335502064974818

Graph of the ZZ-function along the critical line