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Properties

Label 2-28e2-112.3-c1-0-50
Degree 22
Conductor 784784
Sign 0.353−0.935i0.353 - 0.935i
Analytic cond. 6.260276.26027
Root an. cond. 2.502052.50205
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  + (0.655 + 1.25i)2-s + (3.05 − 0.817i)3-s + (−1.14 + 1.64i)4-s + (−0.213 + 0.797i)5-s + (3.02 + 3.28i)6-s + (−2.80 − 0.353i)8-s + (6.05 − 3.49i)9-s + (−1.13 + 0.254i)10-s + (2.73 − 0.732i)11-s + (−2.13 + 5.94i)12-s + (−2.91 + 2.91i)13-s + 2.61i·15-s + (−1.39 − 3.74i)16-s + (2.30 + 1.33i)17-s + (8.34 + 5.29i)18-s + (−0.117 + 0.436i)19-s + ⋯
L(s)  = 1  + (0.463 + 0.886i)2-s + (1.76 − 0.472i)3-s + (−0.570 + 0.821i)4-s + (−0.0956 + 0.356i)5-s + (1.23 + 1.34i)6-s + (−0.992 − 0.125i)8-s + (2.01 − 1.16i)9-s + (−0.360 + 0.0806i)10-s + (0.823 − 0.220i)11-s + (−0.617 + 1.71i)12-s + (−0.807 + 0.807i)13-s + 0.673i·15-s + (−0.348 − 0.937i)16-s + (0.558 + 0.322i)17-s + (1.96 + 1.24i)18-s + (−0.0268 + 0.100i)19-s + ⋯

Functional equation

Λ(s)=(784s/2 ΓC(s) L(s)=((0.353−0.935i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(784s/2 ΓC(s+1/2) L(s)=((0.353−0.935i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 784784    =    24⋅722^{4} \cdot 7^{2}
Sign: 0.353−0.935i0.353 - 0.935i
Analytic conductor: 6.260276.26027
Root analytic conductor: 2.502052.50205
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ784(227,⋅)\chi_{784} (227, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 784, ( :1/2), 0.353−0.935i)(2,\ 784,\ (\ :1/2),\ 0.353 - 0.935i)

Particular Values

L(1)L(1) ≈\approx 2.67546+1.84865i2.67546 + 1.84865i
L(12)L(\frac12) ≈\approx 2.67546+1.84865i2.67546 + 1.84865i
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+(−0.655−1.25i)T 1 + (-0.655 - 1.25i)T
7 1 1
good3 1+(−3.05+0.817i)T+(2.59−1.5i)T2 1 + (-3.05 + 0.817i)T + (2.59 - 1.5i)T^{2}
5 1+(0.213−0.797i)T+(−4.33−2.5i)T2 1 + (0.213 - 0.797i)T + (-4.33 - 2.5i)T^{2}
11 1+(−2.73+0.732i)T+(9.52−5.5i)T2 1 + (-2.73 + 0.732i)T + (9.52 - 5.5i)T^{2}
13 1+(2.91−2.91i)T−13iT2 1 + (2.91 - 2.91i)T - 13iT^{2}
17 1+(−2.30−1.33i)T+(8.5+14.7i)T2 1 + (-2.30 - 1.33i)T + (8.5 + 14.7i)T^{2}
19 1+(0.117−0.436i)T+(−16.4−9.5i)T2 1 + (0.117 - 0.436i)T + (-16.4 - 9.5i)T^{2}
23 1+(−1.63−2.83i)T+(−11.5+19.9i)T2 1 + (-1.63 - 2.83i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.04+2.04i)T+29iT2 1 + (2.04 + 2.04i)T + 29iT^{2}
31 1+(−1.26+2.18i)T+(−15.5−26.8i)T2 1 + (-1.26 + 2.18i)T + (-15.5 - 26.8i)T^{2}
37 1+(2.33+0.625i)T+(32.0+18.5i)T2 1 + (2.33 + 0.625i)T + (32.0 + 18.5i)T^{2}
41 1+11.9T+41T2 1 + 11.9T + 41T^{2}
43 1+(3.27+3.27i)T+43iT2 1 + (3.27 + 3.27i)T + 43iT^{2}
47 1+(4.98+8.63i)T+(−23.5+40.7i)T2 1 + (4.98 + 8.63i)T + (-23.5 + 40.7i)T^{2}
53 1+(0.869+3.24i)T+(−45.8+26.5i)T2 1 + (0.869 + 3.24i)T + (-45.8 + 26.5i)T^{2}
59 1+(1.51+5.66i)T+(−51.0+29.5i)T2 1 + (1.51 + 5.66i)T + (-51.0 + 29.5i)T^{2}
61 1+(6.18+1.65i)T+(52.8+30.5i)T2 1 + (6.18 + 1.65i)T + (52.8 + 30.5i)T^{2}
67 1+(1.60+5.97i)T+(−58.0+33.5i)T2 1 + (1.60 + 5.97i)T + (-58.0 + 33.5i)T^{2}
71 1−5.14T+71T2 1 - 5.14T + 71T^{2}
73 1+(3.49−6.05i)T+(−36.5−63.2i)T2 1 + (3.49 - 6.05i)T + (-36.5 - 63.2i)T^{2}
79 1+(−9.72+5.61i)T+(39.5−68.4i)T2 1 + (-9.72 + 5.61i)T + (39.5 - 68.4i)T^{2}
83 1+(−5.39−5.39i)T+83iT2 1 + (-5.39 - 5.39i)T + 83iT^{2}
89 1+(0.528+0.915i)T+(−44.5+77.0i)T2 1 + (0.528 + 0.915i)T + (-44.5 + 77.0i)T^{2}
97 1+13.2iT−97T2 1 + 13.2iT - 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

−9.959995295142695744237655250370, −9.265410171082717120980103462094, −8.594086561658091873414181309560, −7.82080491097770185107232711152, −7.04347347822399029119576374827, −6.53193081847099797735436500200, −4.99233520148897824732551387867, −3.76477725572442056965658152401, −3.24640534802275837395161340569, −1.89795553094800799942012623000, 1.45968889598379672270362460065, 2.74826104313597079713235838812, 3.36039419733889654201168862106, 4.45015878940255851117670135711, 5.09071771826858533021998121776, 6.74873245297663858571339078590, 7.907095614694205127249017836781, 8.712851658564660791617020411179, 9.343784085276534071298941311211, 10.03061245897545332616232580067

Graph of the ZZ-function along the critical line