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Properties

Label 2-1386-7.4-c1-0-13
Degree 22
Conductor 13861386
Sign 0.0725+0.997i0.0725 + 0.997i
Analytic cond. 11.067211.0672
Root an. cond. 3.326753.32675
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.207 + 0.358i)5-s + (−2.62 + 0.358i)7-s + 0.999·8-s + (0.207 − 0.358i)10-s + (−0.5 + 0.866i)11-s − 1.17·13-s + (1.62 + 2.09i)14-s + (−0.5 − 0.866i)16-s + (1.08 − 1.88i)17-s + (−0.414 − 0.717i)19-s − 0.414·20-s + 0.999·22-s + (1.62 + 2.80i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.0926 + 0.160i)5-s + (−0.990 + 0.135i)7-s + 0.353·8-s + (0.0654 − 0.113i)10-s + (−0.150 + 0.261i)11-s − 0.324·13-s + (0.433 + 0.558i)14-s + (−0.125 − 0.216i)16-s + (0.263 − 0.456i)17-s + (−0.0950 − 0.164i)19-s − 0.0926·20-s + 0.213·22-s + (0.338 + 0.585i)23-s + ⋯

Functional equation

Λ(s)=(1386s/2 ΓC(s) L(s)=((0.0725+0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1386s/2 ΓC(s+1/2) L(s)=((0.0725+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13861386    =    2⋅32⋅7⋅112 \cdot 3^{2} \cdot 7 \cdot 11
Sign: 0.0725+0.997i0.0725 + 0.997i
Analytic conductor: 11.067211.0672
Root analytic conductor: 3.326753.32675
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1386(991,⋅)\chi_{1386} (991, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1386, ( :1/2), 0.0725+0.997i)(2,\ 1386,\ (\ :1/2),\ 0.0725 + 0.997i)

Particular Values

L(1)L(1) ≈\approx 0.99029696480.9902969648
L(12)L(\frac12) ≈\approx 0.99029696480.9902969648
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1 1
7 1+(2.62−0.358i)T 1 + (2.62 - 0.358i)T
11 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good5 1+(−0.207−0.358i)T+(−2.5+4.33i)T2 1 + (-0.207 - 0.358i)T + (-2.5 + 4.33i)T^{2}
13 1+1.17T+13T2 1 + 1.17T + 13T^{2}
17 1+(−1.08+1.88i)T+(−8.5−14.7i)T2 1 + (-1.08 + 1.88i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.414+0.717i)T+(−9.5+16.4i)T2 1 + (0.414 + 0.717i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.62−2.80i)T+(−11.5+19.9i)T2 1 + (-1.62 - 2.80i)T + (-11.5 + 19.9i)T^{2}
29 1−2.82T+29T2 1 - 2.82T + 29T^{2}
31 1+(−3.24+5.61i)T+(−15.5−26.8i)T2 1 + (-3.24 + 5.61i)T + (-15.5 - 26.8i)T^{2}
37 1+(4.82+8.36i)T+(−18.5+32.0i)T2 1 + (4.82 + 8.36i)T + (-18.5 + 32.0i)T^{2}
41 1−4.65T+41T2 1 - 4.65T + 41T^{2}
43 1+2.82T+43T2 1 + 2.82T + 43T^{2}
47 1+(−4.62−8.00i)T+(−23.5+40.7i)T2 1 + (-4.62 - 8.00i)T + (-23.5 + 40.7i)T^{2}
53 1+(−2.58+4.47i)T+(−26.5−45.8i)T2 1 + (-2.58 + 4.47i)T + (-26.5 - 45.8i)T^{2}
59 1+(−1.82+3.16i)T+(−29.5−51.0i)T2 1 + (-1.82 + 3.16i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.792−1.37i)T+(−30.5+52.8i)T2 1 + (-0.792 - 1.37i)T + (-30.5 + 52.8i)T^{2}
67 1+(−6.74+11.6i)T+(−33.5−58.0i)T2 1 + (-6.74 + 11.6i)T + (-33.5 - 58.0i)T^{2}
71 1−13.3T+71T2 1 - 13.3T + 71T^{2}
73 1+(−2.41+4.18i)T+(−36.5−63.2i)T2 1 + (-2.41 + 4.18i)T + (-36.5 - 63.2i)T^{2}
79 1+(2.37+4.11i)T+(−39.5+68.4i)T2 1 + (2.37 + 4.11i)T + (-39.5 + 68.4i)T^{2}
83 1+9.82T+83T2 1 + 9.82T + 83T^{2}
89 1+(6.24+10.8i)T+(−44.5+77.0i)T2 1 + (6.24 + 10.8i)T + (-44.5 + 77.0i)T^{2}
97 1+10.1T+97T2 1 + 10.1T + 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

−9.529818951296537360587761561041, −8.825608951632944417830845977678, −7.80542594501443287958165685185, −7.02897476909572800431686160951, −6.16901182453900406748280458673, −5.11056901136977038405331426200, −4.04172649309179217644802879903, −3.02829206909068152429715690932, −2.24354150718876193339065855400, −0.55407289796744983334991458105, 1.04621127453767368454729911395, 2.70190915678549223913815716721, 3.75773428502498641542591272061, 4.92718593687242269197684456522, 5.73394796241594168143685231731, 6.68078713577740265998364539213, 7.13298533054877379104933178456, 8.340372455464620669009341998868, 8.774630435469290771339669069384, 9.819700189522624510861094223997

Graph of the ZZ-function along the critical line