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Properties

Label 2-38-19.11-c5-0-4
Degree 22
Conductor 3838
Sign 0.981−0.193i0.981 - 0.193i
Analytic cond. 6.094586.09458
Root an. cond. 2.468722.46872
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  + (−2 + 3.46i)2-s + (−3.86 + 6.68i)3-s + (−7.99 − 13.8i)4-s + (46.3 − 80.3i)5-s + (−15.4 − 26.7i)6-s + 13.5·7-s + 63.9·8-s + (91.6 + 158. i)9-s + (185. + 321. i)10-s + 407.·11-s + 123.·12-s + (−86.6 − 150. i)13-s + (−27.0 + 46.9i)14-s + (358. + 620. i)15-s + (−128 + 221. i)16-s + (822. − 1.42e3i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.247 + 0.429i)3-s + (−0.249 − 0.433i)4-s + (0.829 − 1.43i)5-s + (−0.175 − 0.303i)6-s + 0.104·7-s + 0.353·8-s + (0.377 + 0.653i)9-s + (0.586 + 1.01i)10-s + 1.01·11-s + 0.247·12-s + (−0.142 − 0.246i)13-s + (−0.0369 + 0.0639i)14-s + (0.410 + 0.711i)15-s + (−0.125 + 0.216i)16-s + (0.690 − 1.19i)17-s + ⋯

Functional equation

Λ(s)=(38s/2 ΓC(s) L(s)=((0.981−0.193i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(38s/2 ΓC(s+5/2) L(s)=((0.981−0.193i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3838    =    2⋅192 \cdot 19
Sign: 0.981−0.193i0.981 - 0.193i
Analytic conductor: 6.094586.09458
Root analytic conductor: 2.468722.46872
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ38(11,⋅)\chi_{38} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 38, ( :5/2), 0.981−0.193i)(2,\ 38,\ (\ :5/2),\ 0.981 - 0.193i)

Particular Values

L(3)L(3) ≈\approx 1.44871+0.141750i1.44871 + 0.141750i
L(12)L(\frac12) ≈\approx 1.44871+0.141750i1.44871 + 0.141750i
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+(2−3.46i)T 1 + (2 - 3.46i)T
19 1+(−1.57e3−27.9i)T 1 + (-1.57e3 - 27.9i)T
good3 1+(3.86−6.68i)T+(−121.5−210.i)T2 1 + (3.86 - 6.68i)T + (-121.5 - 210. i)T^{2}
5 1+(−46.3+80.3i)T+(−1.56e3−2.70e3i)T2 1 + (-46.3 + 80.3i)T + (-1.56e3 - 2.70e3i)T^{2}
7 1−13.5T+1.68e4T2 1 - 13.5T + 1.68e4T^{2}
11 1−407.T+1.61e5T2 1 - 407.T + 1.61e5T^{2}
13 1+(86.6+150.i)T+(−1.85e5+3.21e5i)T2 1 + (86.6 + 150. i)T + (-1.85e5 + 3.21e5i)T^{2}
17 1+(−822.+1.42e3i)T+(−7.09e5−1.22e6i)T2 1 + (-822. + 1.42e3i)T + (-7.09e5 - 1.22e6i)T^{2}
23 1+(−778.−1.34e3i)T+(−3.21e6+5.57e6i)T2 1 + (-778. - 1.34e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1+(2.20e3+3.81e3i)T+(−1.02e7+1.77e7i)T2 1 + (2.20e3 + 3.81e3i)T + (-1.02e7 + 1.77e7i)T^{2}
31 1+1.48e3T+2.86e7T2 1 + 1.48e3T + 2.86e7T^{2}
37 1−7.11e3T+6.93e7T2 1 - 7.11e3T + 6.93e7T^{2}
41 1+(9.66e3−1.67e4i)T+(−5.79e7−1.00e8i)T2 1 + (9.66e3 - 1.67e4i)T + (-5.79e7 - 1.00e8i)T^{2}
43 1+(−7.84e3+1.35e4i)T+(−7.35e7−1.27e8i)T2 1 + (-7.84e3 + 1.35e4i)T + (-7.35e7 - 1.27e8i)T^{2}
47 1+(−6.51e3−1.12e4i)T+(−1.14e8+1.98e8i)T2 1 + (-6.51e3 - 1.12e4i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−820.−1.42e3i)T+(−2.09e8+3.62e8i)T2 1 + (-820. - 1.42e3i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(2.34e4−4.06e4i)T+(−3.57e8−6.19e8i)T2 1 + (2.34e4 - 4.06e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(1.38e4+2.39e4i)T+(−4.22e8+7.31e8i)T2 1 + (1.38e4 + 2.39e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(6.09e3+1.05e4i)T+(−6.75e8+1.16e9i)T2 1 + (6.09e3 + 1.05e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+(6.07e3−1.05e4i)T+(−9.02e8−1.56e9i)T2 1 + (6.07e3 - 1.05e4i)T + (-9.02e8 - 1.56e9i)T^{2}
73 1+(2.30e4−3.98e4i)T+(−1.03e9−1.79e9i)T2 1 + (2.30e4 - 3.98e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(2.26e4−3.92e4i)T+(−1.53e9−2.66e9i)T2 1 + (2.26e4 - 3.92e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−2.36e4T+3.93e9T2 1 - 2.36e4T + 3.93e9T^{2}
89 1+(5.75e4+9.97e4i)T+(−2.79e9+4.83e9i)T2 1 + (5.75e4 + 9.97e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1+(1.95e4−3.37e4i)T+(−4.29e9−7.43e9i)T2 1 + (1.95e4 - 3.37e4i)T + (-4.29e9 - 7.43e9i)T^{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

−15.71914372812583369668296940453, −14.12182948817379327869309178710, −13.16910691322104839230518168844, −11.67333786209880031910790889578, −9.836512979984287251146414921046, −9.216093812196277248307554844390, −7.64844536032300105947230201479, −5.68927117932890158187300857527, −4.70771868366663079448718224737, −1.21386934800942374400245448098, 1.58961081282473049753404043500, 3.45260822281105648367370282612, 6.16752379674396624988752092214, 7.25661304167949953723065053437, 9.298514749734345819503805984329, 10.34323473321207605132207964334, 11.49491725495303207868512580161, 12.66395796914739456495128145834, 14.09171782027158295510088904513, 14.90669632568257192877313302404

Graph of the ZZ-function along the critical line