Properties

Label 2-366-61.13-c1-0-5
Degree 22
Conductor 366366
Sign 0.255+0.966i0.255 + 0.966i
Analytic cond. 2.922522.92252
Root an. cond. 1.709531.70953
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + (−1.74 + 3.02i)5-s + (−0.5 + 0.866i)6-s + (2.04 − 3.53i)7-s − 0.999·8-s + 9-s + (1.74 + 3.02i)10-s + 4.08·11-s + (0.499 + 0.866i)12-s + (3.08 − 5.33i)13-s + (−2.04 − 3.53i)14-s + (1.74 − 3.02i)15-s + (−0.5 + 0.866i)16-s + (−3.83 − 6.64i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (−0.780 + 1.35i)5-s + (−0.204 + 0.353i)6-s + (0.771 − 1.33i)7-s − 0.353·8-s + 0.333·9-s + (0.551 + 0.955i)10-s + 1.23·11-s + (0.144 + 0.249i)12-s + (0.854 − 1.47i)13-s + (−0.545 − 0.944i)14-s + (0.450 − 0.780i)15-s + (−0.125 + 0.216i)16-s + (−0.930 − 1.61i)17-s + ⋯

Functional equation

Λ(s)=(366s/2ΓC(s)L(s)=((0.255+0.966i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(366s/2ΓC(s+1/2)L(s)=((0.255+0.966i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 366366    =    23612 \cdot 3 \cdot 61
Sign: 0.255+0.966i0.255 + 0.966i
Analytic conductor: 2.922522.92252
Root analytic conductor: 1.709531.70953
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ366(13,)\chi_{366} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 366, ( :1/2), 0.255+0.966i)(2,\ 366,\ (\ :1/2),\ 0.255 + 0.966i)

Particular Values

L(1)L(1) \approx 1.013940.780773i1.01394 - 0.780773i
L(12)L(\frac12) \approx 1.013940.780773i1.01394 - 0.780773i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1+T 1 + T
61 1+(1.88+7.57i)T 1 + (-1.88 + 7.57i)T
good5 1+(1.743.02i)T+(2.54.33i)T2 1 + (1.74 - 3.02i)T + (-2.5 - 4.33i)T^{2}
7 1+(2.04+3.53i)T+(3.56.06i)T2 1 + (-2.04 + 3.53i)T + (-3.5 - 6.06i)T^{2}
11 14.08T+11T2 1 - 4.08T + 11T^{2}
13 1+(3.08+5.33i)T+(6.511.2i)T2 1 + (-3.08 + 5.33i)T + (-6.5 - 11.2i)T^{2}
17 1+(3.83+6.64i)T+(8.5+14.7i)T2 1 + (3.83 + 6.64i)T + (-8.5 + 14.7i)T^{2}
19 1+(11.73i)T+(9.5+16.4i)T2 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2}
23 11.59T+23T2 1 - 1.59T + 23T^{2}
29 1+(4.587.93i)T+(14.5+25.1i)T2 1 + (-4.58 - 7.93i)T + (-14.5 + 25.1i)T^{2}
31 1+(11.73i)T+(15.5+26.8i)T2 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2}
37 1+2.59T+37T2 1 + 2.59T + 37T^{2}
41 17.08T+41T2 1 - 7.08T + 41T^{2}
43 1+(1.282.22i)T+(21.537.2i)T2 1 + (1.28 - 2.22i)T + (-21.5 - 37.2i)T^{2}
47 1+(4.69+8.13i)T+(23.5+40.7i)T2 1 + (4.69 + 8.13i)T + (-23.5 + 40.7i)T^{2}
53 1+6.30T+53T2 1 + 6.30T + 53T^{2}
59 1+(2.49+4.31i)T+(29.551.0i)T2 1 + (-2.49 + 4.31i)T + (-29.5 - 51.0i)T^{2}
67 1+(0.2040.354i)T+(33.558.0i)T2 1 + (0.204 - 0.354i)T + (-33.5 - 58.0i)T^{2}
71 1+(5.088.79i)T+(35.5+61.4i)T2 1 + (-5.08 - 8.79i)T + (-35.5 + 61.4i)T^{2}
73 1+(2.54.33i)T+(36.5+63.2i)T2 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2}
79 1+(4.678.09i)T+(39.568.4i)T2 1 + (4.67 - 8.09i)T + (-39.5 - 68.4i)T^{2}
83 1+(4.046.99i)T+(41.571.8i)T2 1 + (4.04 - 6.99i)T + (-41.5 - 71.8i)T^{2}
89 1+15.8T+89T2 1 + 15.8T + 89T^{2}
97 1+(2.664.61i)T+(48.5+84.0i)T2 1 + (-2.66 - 4.61i)T + (-48.5 + 84.0i)T^{2}
show more
show less
   L(s)=p j=12(1αj,pps)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

−11.06226236907451347449265835146, −10.83440129384430569778741599980, −9.848129090766527381068675983526, −8.315108618785523235353235887839, −7.15110907813220437291870109792, −6.64151957066115084756796554573, −5.07942057319333933364028445653, −3.97692576167411312892350600542, −3.14272014261601278187200170090, −0.985782425999355946798627676445, 1.61121903139439516655386133868, 4.19936781175274643038305336093, 4.52444475357406099646555038591, 5.85041306155827942672045660020, 6.53606426263852018214995564663, 8.065616019997167754424440484929, 8.776017942831535473935181055689, 9.213375654867571259403888807819, 11.25802714273247170837101090097, 11.72056815726889972161083978427

Graph of the ZZ-function along the critical line