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Properties

Label 2-13-13.8-c2-0-0
Degree 22
Conductor 1313
Sign 0.984−0.176i0.984 - 0.176i
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  + (0.581 + 0.581i)2-s − 4.16·3-s − 3.32i·4-s + (3.58 + 3.58i)5-s + (−2.41 − 2.41i)6-s + (−4.58 + 4.58i)7-s + (4.25 − 4.25i)8-s + 8.32·9-s + 4.16i·10-s + (5.32 − 5.32i)11-s + 13.8i·12-s + (−5.90 − 11.5i)13-s − 5.32·14-s + (−14.9 − 14.9i)15-s − 8.35·16-s + 21.9i·17-s + ⋯
L(s)  = 1  + (0.290 + 0.290i)2-s − 1.38·3-s − 0.831i·4-s + (0.716 + 0.716i)5-s + (−0.403 − 0.403i)6-s + (−0.654 + 0.654i)7-s + (0.532 − 0.532i)8-s + 0.924·9-s + 0.416i·10-s + (0.484 − 0.484i)11-s + 1.15i·12-s + (−0.454 − 0.890i)13-s − 0.380·14-s + (−0.993 − 0.993i)15-s − 0.521·16-s + 1.29i·17-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=((0.984−0.176i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+1) L(s)=((0.984−0.176i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: 0.984−0.176i0.984 - 0.176i
Analytic conductor: 0.3542240.354224
Root analytic conductor: 0.5951670.595167
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ13(8,⋅)\chi_{13} (8, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 13, ( :1), 0.984−0.176i)(2,\ 13,\ (\ :1),\ 0.984 - 0.176i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.648809+0.0577555i0.648809 + 0.0577555i
L(12)L(\frac12) ≈\approx 0.648809+0.0577555i0.648809 + 0.0577555i
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+(5.90+11.5i)T 1 + (5.90 + 11.5i)T
good2 1+(−0.581−0.581i)T+4iT2 1 + (-0.581 - 0.581i)T + 4iT^{2}
3 1+4.16T+9T2 1 + 4.16T + 9T^{2}
5 1+(−3.58−3.58i)T+25iT2 1 + (-3.58 - 3.58i)T + 25iT^{2}
7 1+(4.58−4.58i)T−49iT2 1 + (4.58 - 4.58i)T - 49iT^{2}
11 1+(−5.32+5.32i)T−121iT2 1 + (-5.32 + 5.32i)T - 121iT^{2}
17 1−21.9iT−289T2 1 - 21.9iT - 289T^{2}
19 1+(−3.16−3.16i)T+361iT2 1 + (-3.16 - 3.16i)T + 361iT^{2}
23 1−8.51iT−529T2 1 - 8.51iT - 529T^{2}
29 1+5.81T+841T2 1 + 5.81T + 841T^{2}
31 1+(−0.513−0.513i)T+961iT2 1 + (-0.513 - 0.513i)T + 961iT^{2}
37 1+(−24.2+24.2i)T−1.36e3iT2 1 + (-24.2 + 24.2i)T - 1.36e3iT^{2}
41 1+(−4.83−4.83i)T+1.68e3iT2 1 + (-4.83 - 4.83i)T + 1.68e3iT^{2}
43 1+30.4iT−1.84e3T2 1 + 30.4iT - 1.84e3T^{2}
47 1+(37.3−37.3i)T−2.20e3iT2 1 + (37.3 - 37.3i)T - 2.20e3iT^{2}
53 1+35.8T+2.80e3T2 1 + 35.8T + 2.80e3T^{2}
59 1+(−58.2+58.2i)T−3.48e3iT2 1 + (-58.2 + 58.2i)T - 3.48e3iT^{2}
61 1+80.3T+3.72e3T2 1 + 80.3T + 3.72e3T^{2}
67 1+(−39.0−39.0i)T+4.48e3iT2 1 + (-39.0 - 39.0i)T + 4.48e3iT^{2}
71 1+(−91.5−91.5i)T+5.04e3iT2 1 + (-91.5 - 91.5i)T + 5.04e3iT^{2}
73 1+(−31.6+31.6i)T−5.32e3iT2 1 + (-31.6 + 31.6i)T - 5.32e3iT^{2}
79 1+18.7T+6.24e3T2 1 + 18.7T + 6.24e3T^{2}
83 1+(44.6+44.6i)T+6.88e3iT2 1 + (44.6 + 44.6i)T + 6.88e3iT^{2}
89 1+(−8.89+8.89i)T−7.92e3iT2 1 + (-8.89 + 8.89i)T - 7.92e3iT^{2}
97 1+(121.+121.i)T+9.40e3iT2 1 + (121. + 121. i)T + 9.40e3iT^{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

−19.49695197618202841717339595129, −18.33323662465042175742064478334, −17.18608481629438217751192576964, −15.76229596423075253929326431430, −14.45109489363855657383888504907, −12.73477724760375199072324025869, −10.99395777799112031275258853299, −9.896974242426991948391952809843, −6.38261627920717730506117029383, −5.65990377670370487912622172668, 4.77450417013775926355845063463, 6.85874272103586218025205582972, 9.563996724071608323299871578343, 11.43954033197981425354098322200, 12.48767843173433937554344724520, 13.67826964445310405211522983516, 16.51199170187262004701105122947, 16.81981330681481747967995918196, 17.95968323919624174885987371043, 20.07705563370172261207446866316

Graph of the ZZ-function along the critical line