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Properties

Label 2-1197-7.2-c1-0-43
Degree 22
Conductor 11971197
Sign −0.266+0.963i-0.266 + 0.963i
Analytic cond. 9.558099.55809
Root an. cond. 3.091613.09161
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 − 1.73i)2-s + (−0.999 − 1.73i)4-s + (0.5 − 0.866i)5-s + (2 − 1.73i)7-s + (−0.999 − 1.73i)10-s + (2 + 3.46i)11-s + 4·13-s + (−0.999 − 5.19i)14-s + (1.99 − 3.46i)16-s + (1.5 + 2.59i)17-s + (0.5 − 0.866i)19-s − 1.99·20-s + 7.99·22-s + (−1.5 + 2.59i)23-s + (2 + 3.46i)25-s + (4 − 6.92i)26-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.223 − 0.387i)5-s + (0.755 − 0.654i)7-s + (−0.316 − 0.547i)10-s + (0.603 + 1.04i)11-s + 1.10·13-s + (−0.267 − 1.38i)14-s + (0.499 − 0.866i)16-s + (0.363 + 0.630i)17-s + (0.114 − 0.198i)19-s − 0.447·20-s + 1.70·22-s + (−0.312 + 0.541i)23-s + (0.400 + 0.692i)25-s + (0.784 − 1.35i)26-s + ⋯

Functional equation

Λ(s)=(1197s/2 ΓC(s) L(s)=((−0.266+0.963i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1197s/2 ΓC(s+1/2) L(s)=((−0.266+0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11971197    =    32⋅7⋅193^{2} \cdot 7 \cdot 19
Sign: −0.266+0.963i-0.266 + 0.963i
Analytic conductor: 9.558099.55809
Root analytic conductor: 3.091613.09161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1197(856,⋅)\chi_{1197} (856, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1197, ( :1/2), −0.266+0.963i)(2,\ 1197,\ (\ :1/2),\ -0.266 + 0.963i)

Particular Values

L(1)L(1) ≈\approx 3.0339043253.033904325
L(12)L(\frac12) ≈\approx 3.0339043253.033904325
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)
bad3 1 1
7 1+(−2+1.73i)T 1 + (-2 + 1.73i)T
19 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good2 1+(−1+1.73i)T+(−1−1.73i)T2 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2}
5 1+(−0.5+0.866i)T+(−2.5−4.33i)T2 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2}
11 1+(−2−3.46i)T+(−5.5+9.52i)T2 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2}
13 1−4T+13T2 1 - 4T + 13T^{2}
17 1+(−1.5−2.59i)T+(−8.5+14.7i)T2 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2}
23 1+(1.5−2.59i)T+(−11.5−19.9i)T2 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2}
29 1+10T+29T2 1 + 10T + 29T^{2}
31 1+(−15.5+26.8i)T2 1 + (-15.5 + 26.8i)T^{2}
37 1+(−3+5.19i)T+(−18.5−32.0i)T2 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1+7T+43T2 1 + 7T + 43T^{2}
47 1+(−23.5−40.7i)T2 1 + (-23.5 - 40.7i)T^{2}
53 1+(6+10.3i)T+(−26.5+45.8i)T2 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2}
59 1+(−6−10.3i)T+(−29.5+51.0i)T2 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2}
61 1+(5−8.66i)T+(−30.5−52.8i)T2 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2}
67 1+(5+8.66i)T+(−33.5+58.0i)T2 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1+(3+5.19i)T+(−36.5+63.2i)T2 1 + (3 + 5.19i)T + (-36.5 + 63.2i)T^{2}
79 1+(−5+8.66i)T+(−39.5−68.4i)T2 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2}
83 1+3T+83T2 1 + 3T + 83T^{2}
89 1+(7−12.1i)T+(−44.5−77.0i)T2 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2}
97 1+12T+97T2 1 + 12T + 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

−9.748754648552596153722677321849, −8.957629836994008346834429375009, −7.83709052622580173282315018755, −7.11962274433598327916875275569, −5.79367924138190114320782861346, −4.93213569985568165199959255972, −4.06281629421142232561092527243, −3.49398011524995401196443638318, −1.85822919228122230212410417769, −1.36145542621407559053202370703, 1.53557872247746337890156215252, 3.11674797638828561074673624008, 4.13318000854646644873592841197, 5.14814043757902004970648201263, 5.94572643188098069192041258008, 6.36526725542690696131063994600, 7.38952075981755010033804826381, 8.286427150672961414179379640069, 8.738024783280089023136032357369, 9.892786872988371826042123126679

Graph of the ZZ-function along the critical line