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Properties

Label 2-156-12.11-c1-0-13
Degree 22
Conductor 156156
Sign 0.417+0.908i0.417 + 0.908i
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  + (−1.11 − 0.866i)2-s + (1.70 − 0.306i)3-s + (0.500 + 1.93i)4-s − 2.09i·5-s + (−2.17 − 1.13i)6-s + 0.613i·7-s + (1.11 − 2.59i)8-s + (2.81 − 1.04i)9-s + (−1.81 + 2.33i)10-s + (1.44 + 3.14i)12-s + 13-s + (0.531 − 0.686i)14-s + (−0.641 − 3.56i)15-s + (−3.5 + 1.93i)16-s + 2.09i·17-s + (−4.04 − 1.26i)18-s + ⋯
L(s)  = 1  + (−0.790 − 0.612i)2-s + (0.984 − 0.177i)3-s + (0.250 + 0.968i)4-s − 0.935i·5-s + (−0.886 − 0.462i)6-s + 0.231i·7-s + (0.395 − 0.918i)8-s + (0.937 − 0.348i)9-s + (−0.572 + 0.739i)10-s + (0.417 + 0.908i)12-s + 0.277·13-s + (0.142 − 0.183i)14-s + (−0.165 − 0.920i)15-s + (−0.875 + 0.484i)16-s + 0.507i·17-s + (−0.954 − 0.298i)18-s + ⋯

Functional equation

Λ(s)=(156s/2 ΓC(s) L(s)=((0.417+0.908i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(156s/2 ΓC(s+1/2) L(s)=((0.417+0.908i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 156156    =    22⋅3⋅132^{2} \cdot 3 \cdot 13
Sign: 0.417+0.908i0.417 + 0.908i
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ156(131,⋅)\chi_{156} (131, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 156, ( :1/2), 0.417+0.908i)(2,\ 156,\ (\ :1/2),\ 0.417 + 0.908i)

Particular Values

L(1)L(1) ≈\approx 0.898169−0.575724i0.898169 - 0.575724i
L(12)L(\frac12) ≈\approx 0.898169−0.575724i0.898169 - 0.575724i
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+(1.11+0.866i)T 1 + (1.11 + 0.866i)T
3 1+(−1.70+0.306i)T 1 + (-1.70 + 0.306i)T
13 1−T 1 - T
good5 1+2.09iT−5T2 1 + 2.09iT - 5T^{2}
7 1−0.613iT−7T2 1 - 0.613iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
17 1−2.09iT−17T2 1 - 2.09iT - 17T^{2}
19 1+5.29iT−19T2 1 + 5.29iT - 19T^{2}
23 1+6.81T+23T2 1 + 6.81T + 23T^{2}
29 1−6.92iT−29T2 1 - 6.92iT - 29T^{2}
31 1−5.29iT−31T2 1 - 5.29iT - 31T^{2}
37 1+5.62T+37T2 1 + 5.62T + 37T^{2}
41 1−11.1iT−41T2 1 - 11.1iT - 41T^{2}
43 1−11.1iT−43T2 1 - 11.1iT - 43T^{2}
47 1+3.40T+47T2 1 + 3.40T + 47T^{2}
53 1+11.1iT−53T2 1 + 11.1iT - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+5.29iT−67T2 1 + 5.29iT - 67T^{2}
71 1+3.40T+71T2 1 + 3.40T + 71T^{2}
73 1+13.2T+73T2 1 + 13.2T + 73T^{2}
79 1+6.51iT−79T2 1 + 6.51iT - 79T^{2}
83 1−13.6T+83T2 1 - 13.6T + 83T^{2}
89 1+2.74iT−89T2 1 + 2.74iT - 89T^{2}
97 1+5.24T+97T2 1 + 5.24T + 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

−12.76247590744069376185522783374, −11.86663565673852971356491951120, −10.54431261237114090935999956095, −9.449227739613737767024573117678, −8.707058611912045298565740467227, −8.033983999884523542465872988676, −6.72177548097214016342181970540, −4.59374183235862258100938171323, −3.15550880332309876798858489781, −1.55972770750412420638039260477, 2.22317239066231341118762626536, 3.93727542248255481804512273053, 5.86186181935852707966131976504, 7.12640616153609286378873430152, 7.87499901376151318959768612173, 8.927793488187041543519465444116, 10.07177806289127950495129378163, 10.54736388187699643719279342177, 11.96848152711580845856055102844, 13.76620463784534113776377168119

Graph of the ZZ-function along the critical line