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Properties

Label 2-156-156.107-c1-0-0
Degree 22
Conductor 156156
Sign −0.907−0.419i-0.907 - 0.419i
Analytic cond. 1.245661.24566
Root an. cond. 1.116091.11609
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  + (0.707 + 1.22i)2-s + (−1.72 + 0.178i)3-s + (−0.999 + 1.73i)4-s + 2.23i·5-s + (−1.43 − 1.98i)6-s + (−2.73 − 1.58i)7-s − 2.82·8-s + (2.93 − 0.614i)9-s + (−2.73 + 1.58i)10-s + (2.12 + 3.67i)11-s + (1.41 − 3.16i)12-s + (−3.5 + 0.866i)13-s − 4.47i·14-s + (−0.398 − 3.85i)15-s + (−2.00 − 3.46i)16-s + (1.93 + 1.11i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (−0.994 + 0.102i)3-s + (−0.499 + 0.866i)4-s + 0.999i·5-s + (−0.586 − 0.809i)6-s + (−1.03 − 0.597i)7-s − 0.999·8-s + (0.978 − 0.204i)9-s + (−0.866 + 0.500i)10-s + (0.639 + 1.10i)11-s + (0.408 − 0.912i)12-s + (−0.970 + 0.240i)13-s − 1.19i·14-s + (−0.102 − 0.994i)15-s + (−0.500 − 0.866i)16-s + (0.469 + 0.271i)17-s + ⋯

Functional equation

Λ(s)=(156s/2 ΓC(s) L(s)=((−0.907−0.419i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(156s/2 ΓC(s+1/2) L(s)=((−0.907−0.419i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 156156    =    22⋅3⋅132^{2} \cdot 3 \cdot 13
Sign: −0.907−0.419i-0.907 - 0.419i
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ156(107,⋅)\chi_{156} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 156, ( :1/2), −0.907−0.419i)(2,\ 156,\ (\ :1/2),\ -0.907 - 0.419i)

Particular Values

L(1)L(1) ≈\approx 0.180213+0.818649i0.180213 + 0.818649i
L(12)L(\frac12) ≈\approx 0.180213+0.818649i0.180213 + 0.818649i
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+(−0.707−1.22i)T 1 + (-0.707 - 1.22i)T
3 1+(1.72−0.178i)T 1 + (1.72 - 0.178i)T
13 1+(3.5−0.866i)T 1 + (3.5 - 0.866i)T
good5 1−2.23iT−5T2 1 - 2.23iT - 5T^{2}
7 1+(2.73+1.58i)T+(3.5+6.06i)T2 1 + (2.73 + 1.58i)T + (3.5 + 6.06i)T^{2}
11 1+(−2.12−3.67i)T+(−5.5+9.52i)T2 1 + (-2.12 - 3.67i)T + (-5.5 + 9.52i)T^{2}
17 1+(−1.93−1.11i)T+(8.5+14.7i)T2 1 + (-1.93 - 1.11i)T + (8.5 + 14.7i)T^{2}
19 1+(−5.47−3.16i)T+(9.5+16.4i)T2 1 + (-5.47 - 3.16i)T + (9.5 + 16.4i)T^{2}
23 1+(−1.41−2.44i)T+(−11.5+19.9i)T2 1 + (-1.41 - 2.44i)T + (-11.5 + 19.9i)T^{2}
29 1+(−5.80+3.35i)T+(14.5−25.1i)T2 1 + (-5.80 + 3.35i)T + (14.5 - 25.1i)T^{2}
31 1+3.16iT−31T2 1 + 3.16iT - 31T^{2}
37 1+(0.5+0.866i)T+(−18.5+32.0i)T2 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2}
41 1+(9.68−5.59i)T+(20.5−35.5i)T2 1 + (9.68 - 5.59i)T + (20.5 - 35.5i)T^{2}
43 1+(−2.73−1.58i)T+(21.5+37.2i)T2 1 + (-2.73 - 1.58i)T + (21.5 + 37.2i)T^{2}
47 1−2.82T+47T2 1 - 2.82T + 47T^{2}
53 1+2.23iT−53T2 1 + 2.23iT - 53T^{2}
59 1+(−29.5−51.0i)T2 1 + (-29.5 - 51.0i)T^{2}
61 1+(0.5−0.866i)T+(−30.5−52.8i)T2 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2}
67 1+(5.47−3.16i)T+(33.5−58.0i)T2 1 + (5.47 - 3.16i)T + (33.5 - 58.0i)T^{2}
71 1+(1.41−2.44i)T+(−35.5−61.4i)T2 1 + (1.41 - 2.44i)T + (-35.5 - 61.4i)T^{2}
73 1+3T+73T2 1 + 3T + 73T^{2}
79 1+12.6iT−79T2 1 + 12.6iT - 79T^{2}
83 1−9.89T+83T2 1 - 9.89T + 83T^{2}
89 1+(3.87−2.23i)T+(44.5−77.0i)T2 1 + (3.87 - 2.23i)T + (44.5 - 77.0i)T^{2}
97 1+(−8+13.8i)T+(−48.5−84.0i)T2 1 + (-8 + 13.8i)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

−13.41689762235597094771924293875, −12.33530472273834310787695503905, −11.72012451165303557408384715027, −10.09991823742183288136919979510, −9.670590888379472286760873913929, −7.44339454142121572632951242881, −6.94143386963254968350406030603, −6.04508300965237779054165359083, −4.65156705004792126543715859350, −3.39308774681080973759350447726, 0.821068812612578224845769304164, 3.14753407433093125746151939490, 4.85659907882108061838096849537, 5.58374306567673702825515917550, 6.76323390967179705387764369362, 8.822026341216631218685880451010, 9.617324630957235483079609688916, 10.66353094055424409888938575821, 12.00167287587216319765346794641, 12.17859895067792940895915774764

Graph of the ZZ-function along the critical line