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Properties

Label 2-224-56.3-c1-0-1
Degree 22
Conductor 224224
Sign 0.824+0.565i0.824 + 0.565i
Analytic cond. 1.788641.78864
Root an. cond. 1.337401.33740
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  + (−1.18 − 0.686i)3-s + (0.345 + 0.597i)5-s + (2.63 + 0.222i)7-s + (−0.557 − 0.966i)9-s + (1.63 − 2.82i)11-s + 5.27·13-s − 0.947i·15-s + (−2.20 − 1.27i)17-s + (0.484 − 0.279i)19-s + (−2.98 − 2.07i)21-s + (2.50 − 1.44i)23-s + (2.26 − 3.91i)25-s + 5.64i·27-s − 0.444i·29-s + (−4.45 + 7.71i)31-s + ⋯
L(s)  = 1  + (−0.686 − 0.396i)3-s + (0.154 + 0.267i)5-s + (0.996 + 0.0840i)7-s + (−0.185 − 0.322i)9-s + (0.491 − 0.851i)11-s + 1.46·13-s − 0.244i·15-s + (−0.534 − 0.308i)17-s + (0.111 − 0.0642i)19-s + (−0.650 − 0.452i)21-s + (0.522 − 0.301i)23-s + (0.452 − 0.783i)25-s + 1.08i·27-s − 0.0825i·29-s + (−0.799 + 1.38i)31-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((0.824+0.565i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s+1/2) L(s)=((0.824+0.565i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: 0.824+0.565i0.824 + 0.565i
Analytic conductor: 1.788641.78864
Root analytic conductor: 1.337401.33740
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ224(143,⋅)\chi_{224} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 224, ( :1/2), 0.824+0.565i)(2,\ 224,\ (\ :1/2),\ 0.824 + 0.565i)

Particular Values

L(1)L(1) ≈\approx 1.07912−0.334656i1.07912 - 0.334656i
L(12)L(\frac12) ≈\approx 1.07912−0.334656i1.07912 - 0.334656i
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
7 1+(−2.63−0.222i)T 1 + (-2.63 - 0.222i)T
good3 1+(1.18+0.686i)T+(1.5+2.59i)T2 1 + (1.18 + 0.686i)T + (1.5 + 2.59i)T^{2}
5 1+(−0.345−0.597i)T+(−2.5+4.33i)T2 1 + (-0.345 - 0.597i)T + (-2.5 + 4.33i)T^{2}
11 1+(−1.63+2.82i)T+(−5.5−9.52i)T2 1 + (-1.63 + 2.82i)T + (-5.5 - 9.52i)T^{2}
13 1−5.27T+13T2 1 - 5.27T + 13T^{2}
17 1+(2.20+1.27i)T+(8.5+14.7i)T2 1 + (2.20 + 1.27i)T + (8.5 + 14.7i)T^{2}
19 1+(−0.484+0.279i)T+(9.5−16.4i)T2 1 + (-0.484 + 0.279i)T + (9.5 - 16.4i)T^{2}
23 1+(−2.50+1.44i)T+(11.5−19.9i)T2 1 + (-2.50 + 1.44i)T + (11.5 - 19.9i)T^{2}
29 1+0.444iT−29T2 1 + 0.444iT - 29T^{2}
31 1+(4.45−7.71i)T+(−15.5−26.8i)T2 1 + (4.45 - 7.71i)T + (-15.5 - 26.8i)T^{2}
37 1+(6.00−3.46i)T+(18.5−32.0i)T2 1 + (6.00 - 3.46i)T + (18.5 - 32.0i)T^{2}
41 1−9.76iT−41T2 1 - 9.76iT - 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+(2.20+3.81i)T+(−23.5+40.7i)T2 1 + (2.20 + 3.81i)T + (-23.5 + 40.7i)T^{2}
53 1+(8.17+4.71i)T+(26.5+45.8i)T2 1 + (8.17 + 4.71i)T + (26.5 + 45.8i)T^{2}
59 1+(8.59+4.96i)T+(29.5+51.0i)T2 1 + (8.59 + 4.96i)T + (29.5 + 51.0i)T^{2}
61 1+(−5.23−9.06i)T+(−30.5+52.8i)T2 1 + (-5.23 - 9.06i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.45+2.51i)T+(−33.5−58.0i)T2 1 + (-1.45 + 2.51i)T + (-33.5 - 58.0i)T^{2}
71 1−5.29iT−71T2 1 - 5.29iT - 71T^{2}
73 1+(5.28+3.05i)T+(36.5+63.2i)T2 1 + (5.28 + 3.05i)T + (36.5 + 63.2i)T^{2}
79 1+(−5.01+2.89i)T+(39.5−68.4i)T2 1 + (-5.01 + 2.89i)T + (39.5 - 68.4i)T^{2}
83 1+1.83iT−83T2 1 + 1.83iT - 83T^{2}
89 1+(−1.5+0.866i)T+(44.5−77.0i)T2 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2}
97 1+7.42iT−97T2 1 + 7.42iT - 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

−11.91512230254354141775685966496, −11.23912174697751877792044414029, −10.66268298415281464482322908427, −8.977758998631219437355660168777, −8.349355155470935490950910274293, −6.81433071241681710753875949284, −6.11833166406471587875567237431, −4.95028358443376226661601507317, −3.35055767410698625646299740563, −1.28233500450643950769275025321, 1.71565955527870246854056685864, 3.98105212273386391673769879490, 5.02521687769114316873686548696, 5.95483381892363880126132287818, 7.32309131416298131248198924608, 8.492218820577851254705827509311, 9.405443677855213326546324307269, 10.91093929360230159269360771119, 11.02161747637537463065705652416, 12.18804097880267786029076837020

Graph of the ZZ-function along the critical line