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Properties

Label 2-13e2-13.11-c2-0-14
Degree 22
Conductor 169169
Sign 0.490+0.871i0.490 + 0.871i
Analytic cond. 4.604914.60491
Root an. cond. 2.145902.14590
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.429 + 1.60i)2-s + (−0.677 − 1.17i)3-s + (1.07 + 0.620i)4-s + (−1.57 − 1.57i)5-s + (2.17 − 0.582i)6-s + (−3.25 − 12.1i)7-s + (−6.15 + 6.15i)8-s + (3.58 − 6.20i)9-s + (3.20 − 1.84i)10-s + (−7.38 − 1.97i)11-s − 1.67i·12-s + 20.9·14-s + (−0.780 + 2.91i)15-s + (−4.74 − 8.22i)16-s + (−12.9 − 7.49i)17-s + (8.41 + 8.41i)18-s + ⋯
L(s)  = 1  + (−0.214 + 0.802i)2-s + (−0.225 − 0.390i)3-s + (0.268 + 0.155i)4-s + (−0.314 − 0.314i)5-s + (0.362 − 0.0970i)6-s + (−0.465 − 1.73i)7-s + (−0.769 + 0.769i)8-s + (0.398 − 0.689i)9-s + (0.320 − 0.184i)10-s + (−0.671 − 0.179i)11-s − 0.139i·12-s + 1.49·14-s + (−0.0520 + 0.194i)15-s + (−0.296 − 0.514i)16-s + (−0.763 − 0.440i)17-s + (0.467 + 0.467i)18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.490+0.871i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1) L(s)=((0.490+0.871i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.490+0.871i0.490 + 0.871i
Analytic conductor: 4.604914.60491
Root analytic conductor: 2.145902.14590
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ169(89,⋅)\chi_{169} (89, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1), 0.490+0.871i)(2,\ 169,\ (\ :1),\ 0.490 + 0.871i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.844351−0.493455i0.844351 - 0.493455i
L(12)L(\frac12) ≈\approx 0.844351−0.493455i0.844351 - 0.493455i
L(2)L(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)
bad13 1 1
good2 1+(0.429−1.60i)T+(−3.46−2i)T2 1 + (0.429 - 1.60i)T + (-3.46 - 2i)T^{2}
3 1+(0.677+1.17i)T+(−4.5+7.79i)T2 1 + (0.677 + 1.17i)T + (-4.5 + 7.79i)T^{2}
5 1+(1.57+1.57i)T+25iT2 1 + (1.57 + 1.57i)T + 25iT^{2}
7 1+(3.25+12.1i)T+(−42.4+24.5i)T2 1 + (3.25 + 12.1i)T + (-42.4 + 24.5i)T^{2}
11 1+(7.38+1.97i)T+(104.+60.5i)T2 1 + (7.38 + 1.97i)T + (104. + 60.5i)T^{2}
17 1+(12.9+7.49i)T+(144.5+250.i)T2 1 + (12.9 + 7.49i)T + (144.5 + 250. i)T^{2}
19 1+(−21.4+5.75i)T+(312.−180.5i)T2 1 + (-21.4 + 5.75i)T + (312. - 180.5i)T^{2}
23 1+(−21.0+12.1i)T+(264.5−458.i)T2 1 + (-21.0 + 12.1i)T + (264.5 - 458. i)T^{2}
29 1+(−16.0−27.8i)T+(−420.5+728.i)T2 1 + (-16.0 - 27.8i)T + (-420.5 + 728. i)T^{2}
31 1+(−1.13−1.13i)T+961iT2 1 + (-1.13 - 1.13i)T + 961iT^{2}
37 1+(1.10+0.295i)T+(1.18e3+684.5i)T2 1 + (1.10 + 0.295i)T + (1.18e3 + 684.5i)T^{2}
41 1+(6.93−25.8i)T+(−1.45e3−840.5i)T2 1 + (6.93 - 25.8i)T + (-1.45e3 - 840.5i)T^{2}
43 1+(−2.53−1.46i)T+(924.5+1.60e3i)T2 1 + (-2.53 - 1.46i)T + (924.5 + 1.60e3i)T^{2}
47 1+(−28.5+28.5i)T−2.20e3iT2 1 + (-28.5 + 28.5i)T - 2.20e3iT^{2}
53 1+80.4T+2.80e3T2 1 + 80.4T + 2.80e3T^{2}
59 1+(4.35+16.2i)T+(−3.01e3+1.74e3i)T2 1 + (4.35 + 16.2i)T + (-3.01e3 + 1.74e3i)T^{2}
61 1+(10.8−18.7i)T+(−1.86e3−3.22e3i)T2 1 + (10.8 - 18.7i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(3.71−13.8i)T+(−3.88e3−2.24e3i)T2 1 + (3.71 - 13.8i)T + (-3.88e3 - 2.24e3i)T^{2}
71 1+(−4.18+1.12i)T+(4.36e3−2.52e3i)T2 1 + (-4.18 + 1.12i)T + (4.36e3 - 2.52e3i)T^{2}
73 1+(−26.0+26.0i)T−5.32e3iT2 1 + (-26.0 + 26.0i)T - 5.32e3iT^{2}
79 1−36.3T+6.24e3T2 1 - 36.3T + 6.24e3T^{2}
83 1+(−56.7−56.7i)T+6.88e3iT2 1 + (-56.7 - 56.7i)T + 6.88e3iT^{2}
89 1+(−96.2−25.7i)T+(6.85e3+3.96e3i)T2 1 + (-96.2 - 25.7i)T + (6.85e3 + 3.96e3i)T^{2}
97 1+(−154.+41.3i)T+(8.14e3−4.70e3i)T2 1 + (-154. + 41.3i)T + (8.14e3 - 4.70e3i)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

−12.49052899896780840732472850452, −11.40218602167250379295739658913, −10.41998176007422380990944042526, −9.145851545955451827087438283911, −7.85239452834095536468048288994, −7.08038310896101416137196980258, −6.47313626489025798443723697587, −4.78074351432146988123435687198, −3.24632395194187650220116035113, −0.64236263591886949538325001920, 2.12318917712047719708682656756, 3.22698947342690165039551564011, 5.12608814236645429286118130115, 6.16646124448313934307035731995, 7.57982073003006026323539026793, 9.050133571447039601311357674198, 9.831017922338721237099281245903, 10.84222793049700128838719910817, 11.57003855664544966407648524504, 12.42824521758031186723350492596

Graph of the ZZ-function along the critical line