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Properties

Label 2-650-25.16-c1-0-5
Degree 22
Conductor 650650
Sign −0.956+0.292i-0.956 + 0.292i
Analytic cond. 5.190275.19027
Root an. cond. 2.278212.27821
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  + (0.809 + 0.587i)2-s + (0.925 + 2.84i)3-s + (0.309 + 0.951i)4-s + (−1.66 − 1.49i)5-s + (−0.925 + 2.84i)6-s − 3.79·7-s + (−0.309 + 0.951i)8-s + (−4.83 + 3.51i)9-s + (−0.469 − 2.18i)10-s + (0.674 + 0.490i)11-s + (−2.42 + 1.76i)12-s + (0.809 − 0.587i)13-s + (−3.07 − 2.23i)14-s + (2.71 − 6.12i)15-s + (−0.809 + 0.587i)16-s + (−1.23 + 3.81i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.534 + 1.64i)3-s + (0.154 + 0.475i)4-s + (−0.744 − 0.667i)5-s + (−0.377 + 1.16i)6-s − 1.43·7-s + (−0.109 + 0.336i)8-s + (−1.61 + 1.17i)9-s + (−0.148 − 0.691i)10-s + (0.203 + 0.147i)11-s + (−0.699 + 0.508i)12-s + (0.224 − 0.163i)13-s + (−0.820 − 0.596i)14-s + (0.700 − 1.58i)15-s + (−0.202 + 0.146i)16-s + (−0.300 + 0.925i)17-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((−0.956+0.292i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+1/2) L(s)=((−0.956+0.292i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: −0.956+0.292i-0.956 + 0.292i
Analytic conductor: 5.190275.19027
Root analytic conductor: 2.278212.27821
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ650(391,⋅)\chi_{650} (391, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :1/2), −0.956+0.292i)(2,\ 650,\ (\ :1/2),\ -0.956 + 0.292i)

Particular Values

L(1)L(1) ≈\approx 0.188601−1.26024i0.188601 - 1.26024i
L(12)L(\frac12) ≈\approx 0.188601−1.26024i0.188601 - 1.26024i
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)
bad2 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
5 1+(1.66+1.49i)T 1 + (1.66 + 1.49i)T
13 1+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
good3 1+(−0.925−2.84i)T+(−2.42+1.76i)T2 1 + (-0.925 - 2.84i)T + (-2.42 + 1.76i)T^{2}
7 1+3.79T+7T2 1 + 3.79T + 7T^{2}
11 1+(−0.674−0.490i)T+(3.39+10.4i)T2 1 + (-0.674 - 0.490i)T + (3.39 + 10.4i)T^{2}
17 1+(1.23−3.81i)T+(−13.7−9.99i)T2 1 + (1.23 - 3.81i)T + (-13.7 - 9.99i)T^{2}
19 1+(0.471−1.44i)T+(−15.3−11.1i)T2 1 + (0.471 - 1.44i)T + (-15.3 - 11.1i)T^{2}
23 1+(4.15+3.01i)T+(7.10+21.8i)T2 1 + (4.15 + 3.01i)T + (7.10 + 21.8i)T^{2}
29 1+(−2.30−7.09i)T+(−23.4+17.0i)T2 1 + (-2.30 - 7.09i)T + (-23.4 + 17.0i)T^{2}
31 1+(0.254−0.782i)T+(−25.0−18.2i)T2 1 + (0.254 - 0.782i)T + (-25.0 - 18.2i)T^{2}
37 1+(0.288−0.209i)T+(11.4−35.1i)T2 1 + (0.288 - 0.209i)T + (11.4 - 35.1i)T^{2}
41 1+(−4.84+3.52i)T+(12.6−38.9i)T2 1 + (-4.84 + 3.52i)T + (12.6 - 38.9i)T^{2}
43 1−6.99T+43T2 1 - 6.99T + 43T^{2}
47 1+(0.471+1.45i)T+(−38.0+27.6i)T2 1 + (0.471 + 1.45i)T + (-38.0 + 27.6i)T^{2}
53 1+(−3.50−10.7i)T+(−42.8+31.1i)T2 1 + (-3.50 - 10.7i)T + (-42.8 + 31.1i)T^{2}
59 1+(−0.420+0.305i)T+(18.2−56.1i)T2 1 + (-0.420 + 0.305i)T + (18.2 - 56.1i)T^{2}
61 1+(−11.4−8.33i)T+(18.8+58.0i)T2 1 + (-11.4 - 8.33i)T + (18.8 + 58.0i)T^{2}
67 1+(2.74−8.43i)T+(−54.2−39.3i)T2 1 + (2.74 - 8.43i)T + (-54.2 - 39.3i)T^{2}
71 1+(3.24+9.98i)T+(−57.4+41.7i)T2 1 + (3.24 + 9.98i)T + (-57.4 + 41.7i)T^{2}
73 1+(12.1+8.83i)T+(22.5+69.4i)T2 1 + (12.1 + 8.83i)T + (22.5 + 69.4i)T^{2}
79 1+(2.39+7.36i)T+(−63.9+46.4i)T2 1 + (2.39 + 7.36i)T + (-63.9 + 46.4i)T^{2}
83 1+(2.02−6.24i)T+(−67.1−48.7i)T2 1 + (2.02 - 6.24i)T + (-67.1 - 48.7i)T^{2}
89 1+(−11.1−8.10i)T+(27.5+84.6i)T2 1 + (-11.1 - 8.10i)T + (27.5 + 84.6i)T^{2}
97 1+(−5.12−15.7i)T+(−78.4+57.0i)T2 1 + (-5.12 - 15.7i)T + (-78.4 + 57.0i)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

−10.73073240502082675859182920542, −10.20066154079190193836080853446, −9.046170493275370221978982878520, −8.752584615674140906454848386697, −7.63401469801438262658826046919, −6.34188892191475988573333557641, −5.39994606916419375694723823513, −4.21723671496265788476972890598, −3.86542618384766533533676293962, −2.88884822584651429296405490779, 0.51052454950118445178715951548, 2.34006292144564508794835021354, 3.06750634929521965497408927903, 4.02996355370765609533418446898, 5.94321735110715372249069042882, 6.59337699386899201191169202593, 7.21293590729376941550983145851, 8.092974114787699915602154883230, 9.207141763642038394234666848385, 10.09517725561544332889673481138

Graph of the ZZ-function along the critical line