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Properties

Label 2-7e2-49.23-c1-0-1
Degree 22
Conductor 4949
Sign 0.305−0.952i0.305 - 0.952i
Analytic cond. 0.3912660.391266
Root an. cond. 0.6255130.625513
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  + (−0.151 + 2.01i)2-s + (1.33 − 0.412i)3-s + (−2.06 − 0.310i)4-s + (−1.66 − 1.54i)5-s + (0.629 + 2.76i)6-s + (−1.77 − 1.95i)7-s + (0.0388 − 0.170i)8-s + (−0.858 + 0.584i)9-s + (3.37 − 3.12i)10-s + (1.57 + 1.07i)11-s + (−2.88 + 0.435i)12-s + (4.29 − 2.06i)13-s + (4.21 − 3.28i)14-s + (−2.87 − 1.38i)15-s + (−3.65 − 1.12i)16-s + (−1.47 + 3.76i)17-s + ⋯
L(s)  = 1  + (−0.106 + 1.42i)2-s + (0.772 − 0.238i)3-s + (−1.03 − 0.155i)4-s + (−0.745 − 0.692i)5-s + (0.257 + 1.12i)6-s + (−0.671 − 0.740i)7-s + (0.0137 − 0.0601i)8-s + (−0.286 + 0.194i)9-s + (1.06 − 0.989i)10-s + (0.475 + 0.324i)11-s + (−0.834 + 0.125i)12-s + (1.19 − 0.573i)13-s + (1.12 − 0.878i)14-s + (−0.741 − 0.356i)15-s + (−0.912 − 0.281i)16-s + (−0.358 + 0.914i)17-s + ⋯

Functional equation

Λ(s)=(49s/2 ΓC(s) L(s)=((0.305−0.952i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(49s/2 ΓC(s+1/2) L(s)=((0.305−0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4949    =    727^{2}
Sign: 0.305−0.952i0.305 - 0.952i
Analytic conductor: 0.3912660.391266
Root analytic conductor: 0.6255130.625513
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ49(23,⋅)\chi_{49} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 49, ( :1/2), 0.305−0.952i)(2,\ 49,\ (\ :1/2),\ 0.305 - 0.952i)

Particular Values

L(1)L(1) ≈\approx 0.678476+0.494628i0.678476 + 0.494628i
L(12)L(\frac12) ≈\approx 0.678476+0.494628i0.678476 + 0.494628i
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)
bad7 1+(1.77+1.95i)T 1 + (1.77 + 1.95i)T
good2 1+(0.151−2.01i)T+(−1.97−0.298i)T2 1 + (0.151 - 2.01i)T + (-1.97 - 0.298i)T^{2}
3 1+(−1.33+0.412i)T+(2.47−1.68i)T2 1 + (-1.33 + 0.412i)T + (2.47 - 1.68i)T^{2}
5 1+(1.66+1.54i)T+(0.373+4.98i)T2 1 + (1.66 + 1.54i)T + (0.373 + 4.98i)T^{2}
11 1+(−1.57−1.07i)T+(4.01+10.2i)T2 1 + (-1.57 - 1.07i)T + (4.01 + 10.2i)T^{2}
13 1+(−4.29+2.06i)T+(8.10−10.1i)T2 1 + (-4.29 + 2.06i)T + (8.10 - 10.1i)T^{2}
17 1+(1.47−3.76i)T+(−12.4−11.5i)T2 1 + (1.47 - 3.76i)T + (-12.4 - 11.5i)T^{2}
19 1+(−0.218+0.379i)T+(−9.5−16.4i)T2 1 + (-0.218 + 0.379i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.49−6.35i)T+(−16.8+15.6i)T2 1 + (-2.49 - 6.35i)T + (-16.8 + 15.6i)T^{2}
29 1+(5.30+6.64i)T+(−6.45+28.2i)T2 1 + (5.30 + 6.64i)T + (-6.45 + 28.2i)T^{2}
31 1+(0.409+0.709i)T+(−15.5+26.8i)T2 1 + (0.409 + 0.709i)T + (-15.5 + 26.8i)T^{2}
37 1+(−3.68+0.555i)T+(35.3−10.9i)T2 1 + (-3.68 + 0.555i)T + (35.3 - 10.9i)T^{2}
41 1+(−2.40+10.5i)T+(−36.9−17.7i)T2 1 + (-2.40 + 10.5i)T + (-36.9 - 17.7i)T^{2}
43 1+(−1.79−7.87i)T+(−38.7+18.6i)T2 1 + (-1.79 - 7.87i)T + (-38.7 + 18.6i)T^{2}
47 1+(−0.114+1.53i)T+(−46.4−7.00i)T2 1 + (-0.114 + 1.53i)T + (-46.4 - 7.00i)T^{2}
53 1+(−0.818−0.123i)T+(50.6+15.6i)T2 1 + (-0.818 - 0.123i)T + (50.6 + 15.6i)T^{2}
59 1+(2.23−2.07i)T+(4.40−58.8i)T2 1 + (2.23 - 2.07i)T + (4.40 - 58.8i)T^{2}
61 1+(−0.0576+0.00869i)T+(58.2−17.9i)T2 1 + (-0.0576 + 0.00869i)T + (58.2 - 17.9i)T^{2}
67 1+(−6.06−10.5i)T+(−33.5+58.0i)T2 1 + (-6.06 - 10.5i)T + (-33.5 + 58.0i)T^{2}
71 1+(−3.05+3.83i)T+(−15.7−69.2i)T2 1 + (-3.05 + 3.83i)T + (-15.7 - 69.2i)T^{2}
73 1+(0.809+10.8i)T+(−72.1+10.8i)T2 1 + (0.809 + 10.8i)T + (-72.1 + 10.8i)T^{2}
79 1+(1.22−2.13i)T+(−39.5−68.4i)T2 1 + (1.22 - 2.13i)T + (-39.5 - 68.4i)T^{2}
83 1+(4.16+2.00i)T+(51.7+64.8i)T2 1 + (4.16 + 2.00i)T + (51.7 + 64.8i)T^{2}
89 1+(3.35−2.29i)T+(32.5−82.8i)T2 1 + (3.35 - 2.29i)T + (32.5 - 82.8i)T^{2}
97 1−4.05T+97T2 1 - 4.05T + 97T^{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

−15.79744875144760282794257693394, −15.03629688033677944972190542769, −13.76713617058722842780206151260, −13.00829519132183444950190829452, −11.21732687306566565305305467948, −9.226964059605660142590728131912, −8.230176685988074165768637026139, −7.41274009356171215653093201065, −5.91494540484763681950985226682, −3.93237061296174293806989214568, 2.83692731735832490189411058095, 3.75844927420940842911488618835, 6.60715651189536880904590832936, 8.708948973887031283098563696316, 9.412697870493973541493776701858, 10.99492566767919878671855605569, 11.64622543064035633172985139689, 12.89019202858721180510896992985, 14.16515065600748207108430875633, 15.27678747534733236685282038185

Graph of the ZZ-function along the critical line