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Properties

Label 2-29-29.24-c1-0-0
Degree 22
Conductor 2929
Sign 0.357+0.934i0.357 + 0.934i
Analytic cond. 0.2315660.231566
Root an. cond. 0.4812130.481213
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  + (−1.12 − 1.40i)2-s + (−0.0990 − 0.433i)3-s + (−0.277 + 1.21i)4-s + (0.222 + 0.279i)5-s + (−0.500 + 0.626i)6-s + (0.900 + 3.94i)7-s + (−1.22 + 0.588i)8-s + (2.52 − 1.21i)9-s + (0.143 − 0.626i)10-s + (−2.62 − 1.26i)11-s + 0.554·12-s + (−4.67 − 2.25i)13-s + (4.54 − 5.70i)14-s + (0.0990 − 0.124i)15-s + (4.44 + 2.14i)16-s + 1.10·17-s + ⋯
L(s)  = 1  + (−0.794 − 0.996i)2-s + (−0.0571 − 0.250i)3-s + (−0.138 + 0.607i)4-s + (0.0995 + 0.124i)5-s + (−0.204 + 0.255i)6-s + (0.340 + 1.49i)7-s + (−0.432 + 0.208i)8-s + (0.841 − 0.405i)9-s + (0.0452 − 0.198i)10-s + (−0.791 − 0.380i)11-s + 0.160·12-s + (−1.29 − 0.624i)13-s + (1.21 − 1.52i)14-s + (0.0255 − 0.0320i)15-s + (1.11 + 0.535i)16-s + 0.269·17-s + ⋯

Functional equation

Λ(s)=(29s/2 ΓC(s) L(s)=((0.357+0.934i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(29s/2 ΓC(s+1/2) L(s)=((0.357+0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2929
Sign: 0.357+0.934i0.357 + 0.934i
Analytic conductor: 0.2315660.231566
Root analytic conductor: 0.4812130.481213
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ29(24,⋅)\chi_{29} (24, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 29, ( :1/2), 0.357+0.934i)(2,\ 29,\ (\ :1/2),\ 0.357 + 0.934i)

Particular Values

L(1)L(1) ≈\approx 0.411375−0.283092i0.411375 - 0.283092i
L(12)L(\frac12) ≈\approx 0.411375−0.283092i0.411375 - 0.283092i
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)
bad29 1+(−4.38−3.12i)T 1 + (-4.38 - 3.12i)T
good2 1+(1.12+1.40i)T+(−0.445+1.94i)T2 1 + (1.12 + 1.40i)T + (-0.445 + 1.94i)T^{2}
3 1+(0.0990+0.433i)T+(−2.70+1.30i)T2 1 + (0.0990 + 0.433i)T + (-2.70 + 1.30i)T^{2}
5 1+(−0.222−0.279i)T+(−1.11+4.87i)T2 1 + (-0.222 - 0.279i)T + (-1.11 + 4.87i)T^{2}
7 1+(−0.900−3.94i)T+(−6.30+3.03i)T2 1 + (-0.900 - 3.94i)T + (-6.30 + 3.03i)T^{2}
11 1+(2.62+1.26i)T+(6.85+8.60i)T2 1 + (2.62 + 1.26i)T + (6.85 + 8.60i)T^{2}
13 1+(4.67+2.25i)T+(8.10+10.1i)T2 1 + (4.67 + 2.25i)T + (8.10 + 10.1i)T^{2}
17 1−1.10T+17T2 1 - 1.10T + 17T^{2}
19 1+(0.455−1.99i)T+(−17.1−8.24i)T2 1 + (0.455 - 1.99i)T + (-17.1 - 8.24i)T^{2}
23 1+(2.57−3.23i)T+(−5.11−22.4i)T2 1 + (2.57 - 3.23i)T + (-5.11 - 22.4i)T^{2}
31 1+(3.96+4.97i)T+(−6.89+30.2i)T2 1 + (3.96 + 4.97i)T + (-6.89 + 30.2i)T^{2}
37 1+(−2.62+1.26i)T+(23.0−28.9i)T2 1 + (-2.62 + 1.26i)T + (23.0 - 28.9i)T^{2}
41 1−0.396T+41T2 1 - 0.396T + 41T^{2}
43 1+(−3.57+4.48i)T+(−9.56−41.9i)T2 1 + (-3.57 + 4.48i)T + (-9.56 - 41.9i)T^{2}
47 1+(−7.02−3.38i)T+(29.3+36.7i)T2 1 + (-7.02 - 3.38i)T + (29.3 + 36.7i)T^{2}
53 1+(2.71+3.40i)T+(−11.7+51.6i)T2 1 + (2.71 + 3.40i)T + (-11.7 + 51.6i)T^{2}
59 1+9.10T+59T2 1 + 9.10T + 59T^{2}
61 1+(−1.34−5.89i)T+(−54.9+26.4i)T2 1 + (-1.34 - 5.89i)T + (-54.9 + 26.4i)T^{2}
67 1+(0.337−0.162i)T+(41.7−52.3i)T2 1 + (0.337 - 0.162i)T + (41.7 - 52.3i)T^{2}
71 1+(−10.2−4.94i)T+(44.2+55.5i)T2 1 + (-10.2 - 4.94i)T + (44.2 + 55.5i)T^{2}
73 1+(5.57−6.99i)T+(−16.2−71.1i)T2 1 + (5.57 - 6.99i)T + (-16.2 - 71.1i)T^{2}
79 1+(−0.535+0.257i)T+(49.2−61.7i)T2 1 + (-0.535 + 0.257i)T + (49.2 - 61.7i)T^{2}
83 1+(−2.09+9.19i)T+(−74.7−36.0i)T2 1 + (-2.09 + 9.19i)T + (-74.7 - 36.0i)T^{2}
89 1+(−0.887−1.11i)T+(−19.8+86.7i)T2 1 + (-0.887 - 1.11i)T + (-19.8 + 86.7i)T^{2}
97 1+(3.50−15.3i)T+(−87.3−42.0i)T2 1 + (3.50 - 15.3i)T + (-87.3 - 42.0i)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

−17.62016045812302062496678613487, −15.71774688132796024959834704821, −14.68656156434386290435533914760, −12.59967987249435458310029097238, −11.97719977952906570989708688884, −10.45055346583592329462401095693, −9.389528532047503245348854437306, −7.965079999948813456966409270612, −5.66325346560503472403248174041, −2.46259427346092974642579421634, 4.67898505343392185417733475953, 7.03706179140409979728308016066, 7.75962509752452597228196744040, 9.596757290493155621182311957349, 10.57112998500381496376355919585, 12.60491504219486423836120456389, 14.07884477605949240212508926179, 15.36853698529461451713548474764, 16.51348763886378539153465335735, 17.12766126523671246419827101119

Graph of the ZZ-function along the critical line