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Properties

Label 2-224-224.59-c1-0-19
Degree 22
Conductor 224224
Sign −0.641+0.767i-0.641 + 0.767i
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  + (−0.936 − 1.05i)2-s + (0.164 + 0.0216i)3-s + (−0.246 + 1.98i)4-s + (−0.755 + 0.0994i)5-s + (−0.130 − 0.194i)6-s + (−0.876 − 2.49i)7-s + (2.33 − 1.59i)8-s + (−2.87 − 0.769i)9-s + (0.812 + 0.707i)10-s + (2.95 − 2.26i)11-s + (−0.0835 + 0.320i)12-s + (−1.74 − 4.22i)13-s + (−1.82 + 3.26i)14-s − 0.126·15-s + (−3.87 − 0.980i)16-s + (3.44 − 5.96i)17-s + ⋯
L(s)  = 1  + (−0.662 − 0.749i)2-s + (0.0949 + 0.0124i)3-s + (−0.123 + 0.992i)4-s + (−0.337 + 0.0444i)5-s + (−0.0534 − 0.0793i)6-s + (−0.331 − 0.943i)7-s + (0.825 − 0.564i)8-s + (−0.957 − 0.256i)9-s + (0.257 + 0.223i)10-s + (0.890 − 0.683i)11-s + (−0.0241 + 0.0926i)12-s + (−0.485 − 1.17i)13-s + (−0.487 + 0.872i)14-s − 0.0326·15-s + (−0.969 − 0.245i)16-s + (0.835 − 1.44i)17-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((−0.641+0.767i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s+1/2) L(s)=((−0.641+0.767i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: −0.641+0.767i-0.641 + 0.767i
Analytic conductor: 1.788641.78864
Root analytic conductor: 1.337401.33740
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ224(59,⋅)\chi_{224} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 224, ( :1/2), −0.641+0.767i)(2,\ 224,\ (\ :1/2),\ -0.641 + 0.767i)

Particular Values

L(1)L(1) ≈\approx 0.283004−0.605110i0.283004 - 0.605110i
L(12)L(\frac12) ≈\approx 0.283004−0.605110i0.283004 - 0.605110i
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+(0.936+1.05i)T 1 + (0.936 + 1.05i)T
7 1+(0.876+2.49i)T 1 + (0.876 + 2.49i)T
good3 1+(−0.164−0.0216i)T+(2.89+0.776i)T2 1 + (-0.164 - 0.0216i)T + (2.89 + 0.776i)T^{2}
5 1+(0.755−0.0994i)T+(4.82−1.29i)T2 1 + (0.755 - 0.0994i)T + (4.82 - 1.29i)T^{2}
11 1+(−2.95+2.26i)T+(2.84−10.6i)T2 1 + (-2.95 + 2.26i)T + (2.84 - 10.6i)T^{2}
13 1+(1.74+4.22i)T+(−9.19+9.19i)T2 1 + (1.74 + 4.22i)T + (-9.19 + 9.19i)T^{2}
17 1+(−3.44+5.96i)T+(−8.5−14.7i)T2 1 + (-3.44 + 5.96i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.01−0.781i)T+(4.91+18.3i)T2 1 + (-1.01 - 0.781i)T + (4.91 + 18.3i)T^{2}
23 1+(1.63−6.09i)T+(−19.9−11.5i)T2 1 + (1.63 - 6.09i)T + (-19.9 - 11.5i)T^{2}
29 1+(0.478+1.15i)T+(−20.5+20.5i)T2 1 + (0.478 + 1.15i)T + (-20.5 + 20.5i)T^{2}
31 1+(1.64−2.84i)T+(−15.5−26.8i)T2 1 + (1.64 - 2.84i)T + (-15.5 - 26.8i)T^{2}
37 1+(1.24+9.42i)T+(−35.7+9.57i)T2 1 + (1.24 + 9.42i)T + (-35.7 + 9.57i)T^{2}
41 1+(−7.52−7.52i)T+41iT2 1 + (-7.52 - 7.52i)T + 41iT^{2}
43 1+(−5.35−2.21i)T+(30.4+30.4i)T2 1 + (-5.35 - 2.21i)T + (30.4 + 30.4i)T^{2}
47 1+(−4.87+2.81i)T+(23.5−40.7i)T2 1 + (-4.87 + 2.81i)T + (23.5 - 40.7i)T^{2}
53 1+(−1.62−2.11i)T+(−13.7+51.1i)T2 1 + (-1.62 - 2.11i)T + (-13.7 + 51.1i)T^{2}
59 1+(−10.5+8.10i)T+(15.2−56.9i)T2 1 + (-10.5 + 8.10i)T + (15.2 - 56.9i)T^{2}
61 1+(8.78+6.73i)T+(15.7+58.9i)T2 1 + (8.78 + 6.73i)T + (15.7 + 58.9i)T^{2}
67 1+(1.21−9.22i)T+(−64.7−17.3i)T2 1 + (1.21 - 9.22i)T + (-64.7 - 17.3i)T^{2}
71 1+(5.31−5.31i)T−71iT2 1 + (5.31 - 5.31i)T - 71iT^{2}
73 1+(−4.98+1.33i)T+(63.2−36.5i)T2 1 + (-4.98 + 1.33i)T + (63.2 - 36.5i)T^{2}
79 1+(−3.13−5.43i)T+(−39.5+68.4i)T2 1 + (-3.13 - 5.43i)T + (-39.5 + 68.4i)T^{2}
83 1+(−6.31+2.61i)T+(58.6−58.6i)T2 1 + (-6.31 + 2.61i)T + (58.6 - 58.6i)T^{2}
89 1+(11.2+3.01i)T+(77.0+44.5i)T2 1 + (11.2 + 3.01i)T + (77.0 + 44.5i)T^{2}
97 1+0.918iT−97T2 1 + 0.918iT - 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

−11.64874796058923750633432179035, −11.07924182042074022973455951962, −9.863288845464347998506242906744, −9.229306665359626266595840533170, −7.917818436166543243989223412943, −7.30846656359993303102810771828, −5.64794847746080762012968221897, −3.81685445874597456673437774469, −3.02749175176592373859627121654, −0.68788916700983352719206661571, 2.13080289339566821547461511022, 4.24327139375381362257674837525, 5.69627135015551798658925189041, 6.50653695014370182747641951577, 7.73936164704045488116909219329, 8.723818971041789068270487004215, 9.351749412971755442986091541334, 10.44699213700450941630926751094, 11.71609686570200275407748577435, 12.33724059682055262451188810004

Graph of the ZZ-function along the critical line