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Properties

Label 2-13-13.11-c2-0-0
Degree 22
Conductor 1313
Sign 0.846−0.533i0.846 - 0.533i
Analytic cond. 0.3542240.354224
Root an. cond. 0.5951670.595167
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  + (−0.5 + 1.86i)2-s + (−1.36 − 2.36i)3-s + (0.232 + 0.133i)4-s + (−4.36 − 4.36i)5-s + (5.09 − 1.36i)6-s + (2.26 + 8.46i)7-s + (−5.83 + 5.83i)8-s + (0.767 − 1.33i)9-s + (10.3 − 5.96i)10-s + (6.19 + 1.66i)11-s − 0.732i·12-s + (−6.5 − 11.2i)13-s − 16.9·14-s + (−4.36 + 16.2i)15-s + (−7.42 − 12.8i)16-s + (9.99 + 5.76i)17-s + ⋯
L(s)  = 1  + (−0.250 + 0.933i)2-s + (−0.455 − 0.788i)3-s + (0.0580 + 0.0334i)4-s + (−0.873 − 0.873i)5-s + (0.849 − 0.227i)6-s + (0.323 + 1.20i)7-s + (−0.728 + 0.728i)8-s + (0.0853 − 0.147i)9-s + (1.03 − 0.596i)10-s + (0.563 + 0.150i)11-s − 0.0610i·12-s + (−0.5 − 0.866i)13-s − 1.20·14-s + (−0.291 + 1.08i)15-s + (−0.464 − 0.804i)16-s + (0.587 + 0.339i)17-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=((0.846−0.533i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+1) L(s)=((0.846−0.533i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: 0.846−0.533i0.846 - 0.533i
Analytic conductor: 0.3542240.354224
Root analytic conductor: 0.5951670.595167
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ13(11,⋅)\chi_{13} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 13, ( :1), 0.846−0.533i)(2,\ 13,\ (\ :1),\ 0.846 - 0.533i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.586286+0.169291i0.586286 + 0.169291i
L(12)L(\frac12) ≈\approx 0.586286+0.169291i0.586286 + 0.169291i
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+(6.5+11.2i)T 1 + (6.5 + 11.2i)T
good2 1+(0.5−1.86i)T+(−3.46−2i)T2 1 + (0.5 - 1.86i)T + (-3.46 - 2i)T^{2}
3 1+(1.36+2.36i)T+(−4.5+7.79i)T2 1 + (1.36 + 2.36i)T + (-4.5 + 7.79i)T^{2}
5 1+(4.36+4.36i)T+25iT2 1 + (4.36 + 4.36i)T + 25iT^{2}
7 1+(−2.26−8.46i)T+(−42.4+24.5i)T2 1 + (-2.26 - 8.46i)T + (-42.4 + 24.5i)T^{2}
11 1+(−6.19−1.66i)T+(104.+60.5i)T2 1 + (-6.19 - 1.66i)T + (104. + 60.5i)T^{2}
17 1+(−9.99−5.76i)T+(144.5+250.i)T2 1 + (-9.99 - 5.76i)T + (144.5 + 250. i)T^{2}
19 1+(−3.36+0.901i)T+(312.−180.5i)T2 1 + (-3.36 + 0.901i)T + (312. - 180.5i)T^{2}
23 1+(8.49−4.90i)T+(264.5−458.i)T2 1 + (8.49 - 4.90i)T + (264.5 - 458. i)T^{2}
29 1+(−5.69−9.86i)T+(−420.5+728.i)T2 1 + (-5.69 - 9.86i)T + (-420.5 + 728. i)T^{2}
31 1+(−1.92−1.92i)T+961iT2 1 + (-1.92 - 1.92i)T + 961iT^{2}
37 1+(42.1+11.2i)T+(1.18e3+684.5i)T2 1 + (42.1 + 11.2i)T + (1.18e3 + 684.5i)T^{2}
41 1+(−5.08+18.9i)T+(−1.45e3−840.5i)T2 1 + (-5.08 + 18.9i)T + (-1.45e3 - 840.5i)T^{2}
43 1+(−45−25.9i)T+(924.5+1.60e3i)T2 1 + (-45 - 25.9i)T + (924.5 + 1.60e3i)T^{2}
47 1+(−0.320+0.320i)T−2.20e3iT2 1 + (-0.320 + 0.320i)T - 2.20e3iT^{2}
53 1−78.7T+2.80e3T2 1 - 78.7T + 2.80e3T^{2}
59 1+(−10.9−40.9i)T+(−3.01e3+1.74e3i)T2 1 + (-10.9 - 40.9i)T + (-3.01e3 + 1.74e3i)T^{2}
61 1+(49.1−85.1i)T+(−1.86e3−3.22e3i)T2 1 + (49.1 - 85.1i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−19.9+74.5i)T+(−3.88e3−2.24e3i)T2 1 + (-19.9 + 74.5i)T + (-3.88e3 - 2.24e3i)T^{2}
71 1+(31.0−8.31i)T+(4.36e3−2.52e3i)T2 1 + (31.0 - 8.31i)T + (4.36e3 - 2.52e3i)T^{2}
73 1+(−48.2+48.2i)T−5.32e3iT2 1 + (-48.2 + 48.2i)T - 5.32e3iT^{2}
79 1+82.7T+6.24e3T2 1 + 82.7T + 6.24e3T^{2}
83 1+(69.5+69.5i)T+6.88e3iT2 1 + (69.5 + 69.5i)T + 6.88e3iT^{2}
89 1+(31.8+8.52i)T+(6.85e3+3.96e3i)T2 1 + (31.8 + 8.52i)T + (6.85e3 + 3.96e3i)T^{2}
97 1+(−74.8+20.0i)T+(8.14e3−4.70e3i)T2 1 + (-74.8 + 20.0i)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

−19.69946131770991524875045309811, −18.20982450485576413706454723373, −17.22051251081396437134079460275, −15.88555550706980427605586348646, −14.92450749565529743467736844708, −12.35508748671146947038322297182, −11.94096394180814851765994738485, −8.748529320555201240166277877670, −7.49426845231199612039305488387, −5.69553781794938833798720680035, 3.95683416420677932391690554448, 7.14798786949140281813477201047, 9.949203577202307198335670126866, 10.93174628761151014090250123475, 11.79909410586806601335509550978, 14.23948615269701998674708152208, 15.69349632537767292893142502756, 16.93927975372165790455416367734, 18.79531553297321548176777089468, 19.67380957328236826127553915476

Graph of the ZZ-function along the critical line