↑

Properties

Label 2-396-99.43-c0-0-1
Degree 22
Conductor 396396
Sign 0.766+0.642i0.766 + 0.642i
Analytic cond. 0.1976290.197629
Root an. cond. 0.4445550.444555
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + (−1 − 1.73i)5-s + 9-s + (−0.5 + 0.866i)11-s + (−1 − 1.73i)15-s + (0.5 + 0.866i)23-s + (−1.49 + 2.59i)25-s + 27-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s − 37-s + (−1 − 1.73i)45-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s − 53-s + ⋯
L(s)  = 1  + 3-s + (−1 − 1.73i)5-s + 9-s + (−0.5 + 0.866i)11-s + (−1 − 1.73i)15-s + (0.5 + 0.866i)23-s + (−1.49 + 2.59i)25-s + 27-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s − 37-s + (−1 − 1.73i)45-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s − 53-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=((0.766+0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s) L(s)=((0.766+0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: 0.766+0.642i0.766 + 0.642i
Analytic conductor: 0.1976290.197629
Root analytic conductor: 0.4445550.444555
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ396(241,⋅)\chi_{396} (241, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 396, ( :0), 0.766+0.642i)(2,\ 396,\ (\ :0),\ 0.766 + 0.642i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.96814793890.9681479389
L(12)L(\frac12) ≈\approx 0.96814793890.9681479389
L(1)L(1) 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
3 1−T 1 - T
11 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good5 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
7 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
29 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
31 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
37 1+T+T2 1 + T + T^{2}
41 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1+T+T2 1 + T + T^{2}
59 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
71 1+T+T2 1 + T + T^{2}
73 1−T2 1 - T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
89 1+T+T2 1 + T + T^{2}
97 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
show more
show less
   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

−11.70136177925881820698619439201, −10.28648633137896334484492377742, −9.289551231046621611813501617542, −8.662710115179572720583464653053, −7.86856845139893519475827247187, −7.14993318326154384415023470974, −5.19633272524974394269101909565, −4.48479146806682122089604182101, −3.41370750137473105502287963362, −1.60954385196913242968519716196, 2.61582828989042026853766389212, 3.28180609172253435165309953704, 4.33941602508599612843120622822, 6.20915467728156064500234637887, 7.14068872465359179818039593197, 7.87062351680290350812187551406, 8.632733594795348260989589218504, 9.956175888735277583707366660418, 10.71683966024892544806119382673, 11.37822940010574484609253295582

Graph of the ZZ-function along the critical line