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Properties

Label 2-338-13.10-c1-0-8
Degree 22
Conductor 338338
Sign 0.839−0.543i0.839 - 0.543i
Analytic cond. 2.698942.69894
Root an. cond. 1.642841.64284
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  + (0.866 − 0.5i)2-s + (1.5 + 2.59i)3-s + (0.499 − 0.866i)4-s − i·5-s + (2.59 + 1.5i)6-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (−3 + 5.19i)9-s + (−0.5 − 0.866i)10-s + (1.73 − i)11-s + 3·12-s + 0.999·14-s + (2.59 − 1.5i)15-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + 6i·18-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.866 + 1.49i)3-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (1.06 + 0.612i)6-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (−1 + 1.73i)9-s + (−0.158 − 0.273i)10-s + (0.522 − 0.301i)11-s + 0.866·12-s + 0.267·14-s + (0.670 − 0.387i)15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + 1.41i·18-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=((0.839−0.543i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+1/2) L(s)=((0.839−0.543i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: 0.839−0.543i0.839 - 0.543i
Analytic conductor: 2.698942.69894
Root analytic conductor: 1.642841.64284
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ338(23,⋅)\chi_{338} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 338, ( :1/2), 0.839−0.543i)(2,\ 338,\ (\ :1/2),\ 0.839 - 0.543i)

Particular Values

L(1)L(1) ≈\approx 2.28401+0.675586i2.28401 + 0.675586i
L(12)L(\frac12) ≈\approx 2.28401+0.675586i2.28401 + 0.675586i
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
13 1 1
good3 1+(−1.5−2.59i)T+(−1.5+2.59i)T2 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2}
5 1+iT−5T2 1 + iT - 5T^{2}
7 1+(−0.866−0.5i)T+(3.5+6.06i)T2 1 + (-0.866 - 0.5i)T + (3.5 + 6.06i)T^{2}
11 1+(−1.73+i)T+(5.5−9.52i)T2 1 + (-1.73 + i)T + (5.5 - 9.52i)T^{2}
17 1+(1.5−2.59i)T+(−8.5−14.7i)T2 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(5.19+3i)T+(9.5+16.4i)T2 1 + (5.19 + 3i)T + (9.5 + 16.4i)T^{2}
23 1+(2+3.46i)T+(−11.5+19.9i)T2 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2}
29 1+(1+1.73i)T+(−14.5+25.1i)T2 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2}
31 1−4iT−31T2 1 - 4iT - 31T^{2}
37 1+(2.59−1.5i)T+(18.5−32.0i)T2 1 + (2.59 - 1.5i)T + (18.5 - 32.0i)T^{2}
41 1+(20.5−35.5i)T2 1 + (20.5 - 35.5i)T^{2}
43 1+(2.5−4.33i)T+(−21.5−37.2i)T2 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2}
47 1+13iT−47T2 1 + 13iT - 47T^{2}
53 1−12T+53T2 1 - 12T + 53T^{2}
59 1+(8.66+5i)T+(29.5+51.0i)T2 1 + (8.66 + 5i)T + (29.5 + 51.0i)T^{2}
61 1+(−4+6.92i)T+(−30.5−52.8i)T2 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.73−i)T+(33.5−58.0i)T2 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2}
71 1+(−4.33−2.5i)T+(35.5+61.4i)T2 1 + (-4.33 - 2.5i)T + (35.5 + 61.4i)T^{2}
73 1−10iT−73T2 1 - 10iT - 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1+(5.19−3i)T+(44.5−77.0i)T2 1 + (5.19 - 3i)T + (44.5 - 77.0i)T^{2}
97 1+(12.1+7i)T+(48.5+84.0i)T2 1 + (12.1 + 7i)T + (48.5 + 84.0i)T^{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

−11.43911695887120971118537771681, −10.64515890454952294155269271635, −9.913185870502105458090272301155, −8.709542587015666102174173117482, −8.512768303978593408171527276184, −6.60071325138595913732005757206, −5.19701047741107606152159079311, −4.42488536811783229246609145604, −3.57961127437682844819302955625, −2.26087770069826664507901085731, 1.74981381815733341569202639170, 2.91353835114745998091092565214, 4.21412522403204438203093364766, 5.91541864476566554398979113211, 6.82805544314088257573520170084, 7.44753880197435840843448491105, 8.316422503449317614447196602139, 9.256399762484528168308855329567, 10.78814160458541141660259805470, 11.86141300282434313190287599116

Graph of the ZZ-function along the critical line