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Properties

Label 2-338-13.3-c1-0-6
Degree 22
Conductor 338338
Sign 0.522+0.852i0.522 + 0.852i
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  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 3·5-s + (−0.499 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (1 − 1.73i)9-s + (−1.5 − 2.59i)10-s + (3 + 5.19i)11-s + 0.999·12-s + 0.999·14-s + (−1.5 − 2.59i)15-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s − 2·18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 1.34·5-s + (−0.204 + 0.353i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (0.333 − 0.577i)9-s + (−0.474 − 0.821i)10-s + (0.904 + 1.56i)11-s + 0.288·12-s + 0.267·14-s + (−0.387 − 0.670i)15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s − 0.471·18-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=((0.522+0.852i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+1/2) L(s)=((0.522+0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: 0.522+0.852i0.522 + 0.852i
Analytic conductor: 2.698942.69894
Root analytic conductor: 1.642841.64284
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ338(315,⋅)\chi_{338} (315, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 338, ( :1/2), 0.522+0.852i)(2,\ 338,\ (\ :1/2),\ 0.522 + 0.852i)

Particular Values

L(1)L(1) ≈\approx 1.12269−0.629133i1.12269 - 0.629133i
L(12)L(\frac12) ≈\approx 1.12269−0.629133i1.12269 - 0.629133i
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
13 1 1
good3 1+(0.5+0.866i)T+(−1.5+2.59i)T2 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2}
5 1−3T+5T2 1 - 3T + 5T^{2}
7 1+(0.5−0.866i)T+(−3.5−6.06i)T2 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2}
11 1+(−3−5.19i)T+(−5.5+9.52i)T2 1 + (-3 - 5.19i)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+(−1+1.73i)T+(−9.5−16.4i)T2 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1+(3+5.19i)T+(−14.5+25.1i)T2 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+(3.5+6.06i)T+(−18.5+32.0i)T2 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2}
41 1+(−20.5+35.5i)T2 1 + (-20.5 + 35.5i)T^{2}
43 1+(−0.5+0.866i)T+(−21.5−37.2i)T2 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2}
47 1+3T+47T2 1 + 3T + 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+(3−5.19i)T+(−29.5−51.0i)T2 1 + (3 - 5.19i)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+(−7−12.1i)T+(−33.5+58.0i)T2 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2}
71 1+(1.5−2.59i)T+(−35.5−61.4i)T2 1 + (1.5 - 2.59i)T + (-35.5 - 61.4i)T^{2}
73 1+2T+73T2 1 + 2T + 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+12T+83T2 1 + 12T + 83T^{2}
89 1+(3+5.19i)T+(−44.5+77.0i)T2 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2}
97 1+(5−8.66i)T+(−48.5−84.0i)T2 1 + (5 - 8.66i)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

−11.60967619699366416382551611675, −10.19873472295498509785689001808, −9.572584113146321138249905525998, −9.101405691148172230036869748750, −7.42364860981592227254693409780, −6.63631819452094475795831718184, −5.59186003148210604003732412198, −4.20393011018552674614268859520, −2.46865455591262811545435441123, −1.39614105227617374038853612553, 1.51527637428858548459108106953, 3.55887349022475377528249598166, 5.06130526409527960770800136433, 5.92134663573667486756704022253, 6.62572093069655918811370948820, 8.026027357253440522006233339658, 9.001602777616016471459907645130, 9.864935025021312356477139897005, 10.49186441281234235838314060196, 11.35543051478638778090258748762

Graph of the ZZ-function along the critical line