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Properties

Label 2-1216-76.31-c1-0-13
Degree 22
Conductor 12161216
Sign 0.974−0.222i0.974 - 0.222i
Analytic cond. 9.709809.70980
Root an. cond. 3.116053.11605
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  + (0.305 + 0.528i)3-s + (−1.59 − 2.75i)5-s + 2.36i·7-s + (1.31 − 2.27i)9-s + 5.46i·11-s + (2.31 + 1.33i)13-s + (0.971 − 1.68i)15-s + (−0.552 − 0.957i)17-s + (−1.37 − 4.13i)19-s + (−1.24 + 0.720i)21-s + (2.46 + 1.42i)23-s + (−2.57 + 4.46i)25-s + 3.43·27-s + (5.63 + 3.25i)29-s + 1.01·31-s + ⋯
L(s)  = 1  + (0.176 + 0.305i)3-s + (−0.712 − 1.23i)5-s + 0.893i·7-s + (0.437 − 0.758i)9-s + 1.64i·11-s + (0.643 + 0.371i)13-s + (0.250 − 0.434i)15-s + (−0.134 − 0.232i)17-s + (−0.316 − 0.948i)19-s + (−0.272 + 0.157i)21-s + (0.513 + 0.296i)23-s + (−0.514 + 0.892i)25-s + 0.660·27-s + (1.04 + 0.604i)29-s + 0.182·31-s + ⋯

Functional equation

Λ(s)=(1216s/2 ΓC(s) L(s)=((0.974−0.222i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1216s/2 ΓC(s+1/2) L(s)=((0.974−0.222i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12161216    =    26⋅192^{6} \cdot 19
Sign: 0.974−0.222i0.974 - 0.222i
Analytic conductor: 9.709809.70980
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1216(639,⋅)\chi_{1216} (639, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1216, ( :1/2), 0.974−0.222i)(2,\ 1216,\ (\ :1/2),\ 0.974 - 0.222i)

Particular Values

L(1)L(1) ≈\approx 1.6132276371.613227637
L(12)L(\frac12) ≈\approx 1.6132276371.613227637
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
19 1+(1.37+4.13i)T 1 + (1.37 + 4.13i)T
good3 1+(−0.305−0.528i)T+(−1.5+2.59i)T2 1 + (-0.305 - 0.528i)T + (-1.5 + 2.59i)T^{2}
5 1+(1.59+2.75i)T+(−2.5+4.33i)T2 1 + (1.59 + 2.75i)T + (-2.5 + 4.33i)T^{2}
7 1−2.36iT−7T2 1 - 2.36iT - 7T^{2}
11 1−5.46iT−11T2 1 - 5.46iT - 11T^{2}
13 1+(−2.31−1.33i)T+(6.5+11.2i)T2 1 + (-2.31 - 1.33i)T + (6.5 + 11.2i)T^{2}
17 1+(0.552+0.957i)T+(−8.5+14.7i)T2 1 + (0.552 + 0.957i)T + (-8.5 + 14.7i)T^{2}
23 1+(−2.46−1.42i)T+(11.5+19.9i)T2 1 + (-2.46 - 1.42i)T + (11.5 + 19.9i)T^{2}
29 1+(−5.63−3.25i)T+(14.5+25.1i)T2 1 + (-5.63 - 3.25i)T + (14.5 + 25.1i)T^{2}
31 1−1.01T+31T2 1 - 1.01T + 31T^{2}
37 1−0.450iT−37T2 1 - 0.450iT - 37T^{2}
41 1+(−0.336+0.194i)T+(20.5−35.5i)T2 1 + (-0.336 + 0.194i)T + (20.5 - 35.5i)T^{2}
43 1+(−4.96+2.86i)T+(21.5−37.2i)T2 1 + (-4.96 + 2.86i)T + (21.5 - 37.2i)T^{2}
47 1+(−2.91−1.68i)T+(23.5+40.7i)T2 1 + (-2.91 - 1.68i)T + (23.5 + 40.7i)T^{2}
53 1+(−3.53−2.03i)T+(26.5+45.8i)T2 1 + (-3.53 - 2.03i)T + (26.5 + 45.8i)T^{2}
59 1+(−6.82−11.8i)T+(−29.5+51.0i)T2 1 + (-6.82 - 11.8i)T + (-29.5 + 51.0i)T^{2}
61 1+(−6.77+11.7i)T+(−30.5−52.8i)T2 1 + (-6.77 + 11.7i)T + (-30.5 - 52.8i)T^{2}
67 1+(−4.27+7.39i)T+(−33.5−58.0i)T2 1 + (-4.27 + 7.39i)T + (-33.5 - 58.0i)T^{2}
71 1+(−1.07−1.86i)T+(−35.5+61.4i)T2 1 + (-1.07 - 1.86i)T + (-35.5 + 61.4i)T^{2}
73 1+(−3.91−6.78i)T+(−36.5+63.2i)T2 1 + (-3.91 - 6.78i)T + (-36.5 + 63.2i)T^{2}
79 1+(5.57+9.65i)T+(−39.5+68.4i)T2 1 + (5.57 + 9.65i)T + (-39.5 + 68.4i)T^{2}
83 1−4.14iT−83T2 1 - 4.14iT - 83T^{2}
89 1+(−4.19−2.41i)T+(44.5+77.0i)T2 1 + (-4.19 - 2.41i)T + (44.5 + 77.0i)T^{2}
97 1+(0.641−0.370i)T+(48.5−84.0i)T2 1 + (0.641 - 0.370i)T + (48.5 - 84.0i)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

−9.406494048585345217755116736007, −9.040383673355449493566471334869, −8.406421134717823629249745723439, −7.28640376174527422467256990870, −6.55772624891666170416023108940, −5.22757972099651206888870742811, −4.57627878100346871250014223896, −3.88352876759695211646979256546, −2.45180527208939842868713003167, −1.06039411880304173487539663637, 0.907202853264175129552706536982, 2.59522486721728578693300738131, 3.52422571431006747144923502725, 4.20588463126734674923052319824, 5.67599942130219062127587902708, 6.56828564491659444012974343453, 7.23385594805841599856045740109, 8.128016120950141266878141323846, 8.447133588618260757867622428812, 10.02312613072430396577888081315

Graph of the ZZ-function along the critical line