↑

Properties

Label 2-308-7.2-c1-0-1
Degree 22
Conductor 308308
Sign −0.0996−0.995i-0.0996 - 0.995i
Analytic cond. 2.459392.45939
Root an. cond. 1.568241.56824
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  + (0.794 + 1.37i)3-s + (−1.64 + 2.84i)5-s + (2.64 − 0.0963i)7-s + (0.238 − 0.413i)9-s + (0.5 + 0.866i)11-s − 4.98·13-s − 5.22·15-s + (1.84 + 3.20i)17-s + (−2.84 + 4.93i)19-s + (2.23 + 3.56i)21-s + (−0.349 + 0.605i)23-s + (−2.90 − 5.03i)25-s + 5.52·27-s + 7.68·29-s + (−5.25 − 9.10i)31-s + ⋯
L(s)  = 1  + (0.458 + 0.794i)3-s + (−0.735 + 1.27i)5-s + (0.999 − 0.0364i)7-s + (0.0795 − 0.137i)9-s + (0.150 + 0.261i)11-s − 1.38·13-s − 1.34·15-s + (0.448 + 0.777i)17-s + (−0.653 + 1.13i)19-s + (0.487 + 0.776i)21-s + (−0.0729 + 0.126i)23-s + (−0.581 − 1.00i)25-s + 1.06·27-s + 1.42·29-s + (−0.943 − 1.63i)31-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((−0.0996−0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0996 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+1/2) L(s)=((−0.0996−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0996 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: −0.0996−0.995i-0.0996 - 0.995i
Analytic conductor: 2.459392.45939
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ308(177,⋅)\chi_{308} (177, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1/2), −0.0996−0.995i)(2,\ 308,\ (\ :1/2),\ -0.0996 - 0.995i)

Particular Values

L(1)L(1) ≈\approx 0.936114+1.03453i0.936114 + 1.03453i
L(12)L(\frac12) ≈\approx 0.936114+1.03453i0.936114 + 1.03453i
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
7 1+(−2.64+0.0963i)T 1 + (-2.64 + 0.0963i)T
11 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good3 1+(−0.794−1.37i)T+(−1.5+2.59i)T2 1 + (-0.794 - 1.37i)T + (-1.5 + 2.59i)T^{2}
5 1+(1.64−2.84i)T+(−2.5−4.33i)T2 1 + (1.64 - 2.84i)T + (-2.5 - 4.33i)T^{2}
13 1+4.98T+13T2 1 + 4.98T + 13T^{2}
17 1+(−1.84−3.20i)T+(−8.5+14.7i)T2 1 + (-1.84 - 3.20i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.84−4.93i)T+(−9.5−16.4i)T2 1 + (2.84 - 4.93i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.349−0.605i)T+(−11.5−19.9i)T2 1 + (0.349 - 0.605i)T + (-11.5 - 19.9i)T^{2}
29 1−7.68T+29T2 1 - 7.68T + 29T^{2}
31 1+(5.25+9.10i)T+(−15.5+26.8i)T2 1 + (5.25 + 9.10i)T + (-15.5 + 26.8i)T^{2}
37 1+(−5.28+9.15i)T+(−18.5−32.0i)T2 1 + (-5.28 + 9.15i)T + (-18.5 - 32.0i)T^{2}
41 1−7.81T+41T2 1 - 7.81T + 41T^{2}
43 1−1.63T+43T2 1 - 1.63T + 43T^{2}
47 1+(1.15−1.99i)T+(−23.5−40.7i)T2 1 + (1.15 - 1.99i)T + (-23.5 - 40.7i)T^{2}
53 1+(−1.60−2.78i)T+(−26.5+45.8i)T2 1 + (-1.60 - 2.78i)T + (-26.5 + 45.8i)T^{2}
59 1+(3.88+6.72i)T+(−29.5+51.0i)T2 1 + (3.88 + 6.72i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.47−4.29i)T+(−30.5−52.8i)T2 1 + (2.47 - 4.29i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.810−1.40i)T+(−33.5+58.0i)T2 1 + (-0.810 - 1.40i)T + (-33.5 + 58.0i)T^{2}
71 1+2.30T+71T2 1 + 2.30T + 71T^{2}
73 1+(2.70+4.68i)T+(−36.5+63.2i)T2 1 + (2.70 + 4.68i)T + (-36.5 + 63.2i)T^{2}
79 1+(−1.36+2.36i)T+(−39.5−68.4i)T2 1 + (-1.36 + 2.36i)T + (-39.5 - 68.4i)T^{2}
83 1−6.65T+83T2 1 - 6.65T + 83T^{2}
89 1+(−6.43+11.1i)T+(−44.5−77.0i)T2 1 + (-6.43 + 11.1i)T + (-44.5 - 77.0i)T^{2}
97 1−18.6T+97T2 1 - 18.6T + 97T^{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

−11.83854016422797573803060946047, −10.80673204150653615773195454113, −10.24171224551235887943590986490, −9.262791785899758622974151932251, −7.929928471919498375446508986556, −7.43301893057186764375547676831, −6.07477235983964078624930098792, −4.48123149648900932946715113191, −3.76968095752694427187523989997, −2.40265256290060261019015024870, 1.07083556439729716118656628580, 2.59786165038871904367700510929, 4.61233893173603829153451800146, 5.00566227729328721009000076616, 6.92401275326877987531542536111, 7.77785775276031277739374324995, 8.410501946090070817279948154678, 9.231844452979391084434663251360, 10.64052019105558136447448636790, 11.82611521227536084564425589218

Graph of the ZZ-function along the critical line