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Properties

Label 2-338-13.3-c1-0-5
Degree 22
Conductor 338338
Sign −0.522−0.852i-0.522 - 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 + (1.5 + 2.59i)3-s + (−0.499 + 0.866i)4-s + 5-s + (−1.5 + 2.59i)6-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−3 + 5.19i)9-s + (0.5 + 0.866i)10-s + (−1 − 1.73i)11-s − 3·12-s + 0.999·14-s + (1.5 + 2.59i)15-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s − 6·18-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.866 + 1.49i)3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (−0.612 + 1.06i)6-s + (0.188 − 0.327i)7-s − 0.353·8-s + (−1 + 1.73i)9-s + (0.158 + 0.273i)10-s + (−0.301 − 0.522i)11-s − 0.866·12-s + 0.267·14-s + (0.387 + 0.670i)15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s − 1.41·18-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: −0.522−0.852i-0.522 - 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 1.02078+1.82160i1.02078 + 1.82160i
L(12)L(\frac12) ≈\approx 1.02078+1.82160i1.02078 + 1.82160i
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+(−1.5−2.59i)T+(−1.5+2.59i)T2 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2}
5 1−T+5T2 1 - T + 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+(1+1.73i)T+(−5.5+9.52i)T2 1 + (1 + 1.73i)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+(−3+5.19i)T+(−9.5−16.4i)T2 1 + (-3 + 5.19i)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+4T+31T2 1 + 4T + 31T^{2}
37 1+(−1.5−2.59i)T+(−18.5+32.0i)T2 1 + (-1.5 - 2.59i)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+13T+47T2 1 + 13T + 47T^{2}
53 1−12T+53T2 1 - 12T + 53T^{2}
59 1+(5−8.66i)T+(−29.5−51.0i)T2 1 + (5 - 8.66i)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+1.73i)T+(−33.5+58.0i)T2 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2}
71 1+(2.5−4.33i)T+(−35.5−61.4i)T2 1 + (2.5 - 4.33i)T + (-35.5 - 61.4i)T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1+83T2 1 + 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+(−7+12.1i)T+(−48.5−84.0i)T2 1 + (-7 + 12.1i)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

−11.64355721635335982740632901120, −10.77899740582499712534371250157, −9.704886083943442512656515031659, −9.209112803371729887947257938045, −8.192547814175295956492418782356, −7.23163919789772636519910985104, −5.62261035511708516711623105950, −4.89321223784464746283057464781, −3.76962507835140844885983781499, −2.78493287904463981493756497039, 1.50951539290733269938389551045, 2.38435336386728289082888716944, 3.63021027940924732879525151077, 5.40768158342631322031597599700, 6.38893399675427216344801348518, 7.53667461946666212315676486900, 8.319660007028431413684904528859, 9.337977912137359312454990147230, 10.28165707151217771111895206168, 11.59894487277404434885648120520

Graph of the ZZ-function along the critical line