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Properties

Label 2-13-13.8-c2-0-1
Degree 22
Conductor 1313
Sign 0.399+0.916i0.399 + 0.916i
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  + (−2.58 − 2.58i)2-s + 2.16·3-s + 9.32i·4-s + (0.418 + 0.418i)5-s + (−5.58 − 5.58i)6-s + (−1.41 + 1.41i)7-s + (13.7 − 13.7i)8-s − 4.32·9-s − 2.16i·10-s + (−7.32 + 7.32i)11-s + 20.1i·12-s + (9.90 − 8.41i)13-s + 7.32·14-s + (0.905 + 0.905i)15-s − 33.6·16-s − 15.9i·17-s + ⋯
L(s)  = 1  + (−1.29 − 1.29i)2-s + 0.720·3-s + 2.33i·4-s + (0.0837 + 0.0837i)5-s + (−0.930 − 0.930i)6-s + (−0.202 + 0.202i)7-s + (1.71 − 1.71i)8-s − 0.480·9-s − 0.216i·10-s + (−0.665 + 0.665i)11-s + 1.68i·12-s + (0.761 − 0.647i)13-s + 0.523·14-s + (0.0603 + 0.0603i)15-s − 2.10·16-s − 0.939i·17-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=((0.399+0.916i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+1) L(s)=((0.399+0.916i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: 0.399+0.916i0.399 + 0.916i
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), 0.399+0.916i)(2,\ 13,\ (\ :1),\ 0.399 + 0.916i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.424951−0.278524i0.424951 - 0.278524i
L(12)L(\frac12) ≈\approx 0.424951−0.278524i0.424951 - 0.278524i
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+(−9.90+8.41i)T 1 + (-9.90 + 8.41i)T
good2 1+(2.58+2.58i)T+4iT2 1 + (2.58 + 2.58i)T + 4iT^{2}
3 1−2.16T+9T2 1 - 2.16T + 9T^{2}
5 1+(−0.418−0.418i)T+25iT2 1 + (-0.418 - 0.418i)T + 25iT^{2}
7 1+(1.41−1.41i)T−49iT2 1 + (1.41 - 1.41i)T - 49iT^{2}
11 1+(7.32−7.32i)T−121iT2 1 + (7.32 - 7.32i)T - 121iT^{2}
17 1+15.9iT−289T2 1 + 15.9iT - 289T^{2}
19 1+(3.16+3.16i)T+361iT2 1 + (3.16 + 3.16i)T + 361iT^{2}
23 1−27.4iT−529T2 1 - 27.4iT - 529T^{2}
29 1−25.8T+841T2 1 - 25.8T + 841T^{2}
31 1+(−19.4−19.4i)T+961iT2 1 + (-19.4 - 19.4i)T + 961iT^{2}
37 1+(4.23−4.23i)T−1.36e3iT2 1 + (4.23 - 4.23i)T - 1.36e3iT^{2}
41 1+(−11.1−11.1i)T+1.68e3iT2 1 + (-11.1 - 11.1i)T + 1.68e3iT^{2}
43 1+11.5iT−1.84e3T2 1 + 11.5iT - 1.84e3T^{2}
47 1+(−35.3+35.3i)T−2.20e3iT2 1 + (-35.3 + 35.3i)T - 2.20e3iT^{2}
53 1+4.18T+2.80e3T2 1 + 4.18T + 2.80e3T^{2}
59 1+(30.2−30.2i)T−3.48e3iT2 1 + (30.2 - 30.2i)T - 3.48e3iT^{2}
61 1+67.6T+3.72e3T2 1 + 67.6T + 3.72e3T^{2}
67 1+(81.0+81.0i)T+4.48e3iT2 1 + (81.0 + 81.0i)T + 4.48e3iT^{2}
71 1+(−50.4−50.4i)T+5.04e3iT2 1 + (-50.4 - 50.4i)T + 5.04e3iT^{2}
73 1+(31.6−31.6i)T−5.32e3iT2 1 + (31.6 - 31.6i)T - 5.32e3iT^{2}
79 1−50.7T+6.24e3T2 1 - 50.7T + 6.24e3T^{2}
83 1+(−18.6−18.6i)T+6.88e3iT2 1 + (-18.6 - 18.6i)T + 6.88e3iT^{2}
89 1+(−91.1+91.1i)T−7.92e3iT2 1 + (-91.1 + 91.1i)T - 7.92e3iT^{2}
97 1+(−87.3−87.3i)T+9.40e3iT2 1 + (-87.3 - 87.3i)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

−19.69911499565106552817912694077, −18.41613043880210366164494295136, −17.51135788379122017091526354026, −15.73267954907706978465117940994, −13.56268633963231116245228884102, −12.03463667937509193806138129804, −10.50876836275333162196408802027, −9.177307123611610263205322290920, −7.921204180954394322383080744709, −2.85218477777242428197479268848, 6.16404177135625374340439059988, 8.056957803555827474452859535269, 8.998398593171744386351339134601, 10.65636187705288183345272844949, 13.69557914963411258049827622329, 14.90831937210881891886498767921, 16.17479581892520589519357467793, 17.20522381704729022489342949945, 18.65606451794125737962648612962, 19.45841386238803546328531060040

Graph of the ZZ-function along the critical line