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Properties

Label 2-1098-183.101-c1-0-9
Degree 22
Conductor 10981098
Sign 0.986+0.165i0.986 + 0.165i
Analytic cond. 8.767578.76757
Root an. cond. 2.961002.96100
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.0820 + 0.142i)5-s + (−0.594 − 2.22i)7-s + (−0.707 + 0.707i)8-s + (0.0424 − 0.158i)10-s + (−2.55 + 2.55i)11-s + (−0.513 + 0.888i)13-s + (1.14 + 1.99i)14-s + (0.500 − 0.866i)16-s + (2.00 + 0.535i)17-s + (3.04 − 1.75i)19-s + 0.164i·20-s + (1.80 − 3.12i)22-s + (1.10 + 1.10i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (−0.0367 + 0.0635i)5-s + (−0.224 − 0.839i)7-s + (−0.249 + 0.249i)8-s + (0.0134 − 0.0501i)10-s + (−0.769 + 0.769i)11-s + (−0.142 + 0.246i)13-s + (0.307 + 0.532i)14-s + (0.125 − 0.216i)16-s + (0.485 + 0.129i)17-s + (0.697 − 0.402i)19-s + 0.0367i·20-s + (0.384 − 0.666i)22-s + (0.230 + 0.230i)23-s + ⋯

Functional equation

Λ(s)=(1098s/2 ΓC(s) L(s)=((0.986+0.165i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1098s/2 ΓC(s+1/2) L(s)=((0.986+0.165i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10981098    =    2⋅32⋅612 \cdot 3^{2} \cdot 61
Sign: 0.986+0.165i0.986 + 0.165i
Analytic conductor: 8.767578.76757
Root analytic conductor: 2.961002.96100
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1098(467,⋅)\chi_{1098} (467, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1098, ( :1/2), 0.986+0.165i)(2,\ 1098,\ (\ :1/2),\ 0.986 + 0.165i)

Particular Values

L(1)L(1) ≈\approx 1.0708636441.070863644
L(12)L(\frac12) ≈\approx 1.0708636441.070863644
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+(0.965−0.258i)T 1 + (0.965 - 0.258i)T
3 1 1
61 1+(−7.79−0.469i)T 1 + (-7.79 - 0.469i)T
good5 1+(0.0820−0.142i)T+(−2.5−4.33i)T2 1 + (0.0820 - 0.142i)T + (-2.5 - 4.33i)T^{2}
7 1+(0.594+2.22i)T+(−6.06+3.5i)T2 1 + (0.594 + 2.22i)T + (-6.06 + 3.5i)T^{2}
11 1+(2.55−2.55i)T−11iT2 1 + (2.55 - 2.55i)T - 11iT^{2}
13 1+(0.513−0.888i)T+(−6.5−11.2i)T2 1 + (0.513 - 0.888i)T + (-6.5 - 11.2i)T^{2}
17 1+(−2.00−0.535i)T+(14.7+8.5i)T2 1 + (-2.00 - 0.535i)T + (14.7 + 8.5i)T^{2}
19 1+(−3.04+1.75i)T+(9.5−16.4i)T2 1 + (-3.04 + 1.75i)T + (9.5 - 16.4i)T^{2}
23 1+(−1.10−1.10i)T+23iT2 1 + (-1.10 - 1.10i)T + 23iT^{2}
29 1+(−9.16−2.45i)T+(25.1+14.5i)T2 1 + (-9.16 - 2.45i)T + (25.1 + 14.5i)T^{2}
31 1+(−2.47+9.22i)T+(−26.8−15.5i)T2 1 + (-2.47 + 9.22i)T + (-26.8 - 15.5i)T^{2}
37 1+(4.12+4.12i)T+37iT2 1 + (4.12 + 4.12i)T + 37iT^{2}
41 1+8.62T+41T2 1 + 8.62T + 41T^{2}
43 1+(−7.44+1.99i)T+(37.2−21.5i)T2 1 + (-7.44 + 1.99i)T + (37.2 - 21.5i)T^{2}
47 1+(−7.88+4.55i)T+(23.5−40.7i)T2 1 + (-7.88 + 4.55i)T + (23.5 - 40.7i)T^{2}
53 1+(1.12+1.12i)T+53iT2 1 + (1.12 + 1.12i)T + 53iT^{2}
59 1+(1.19+4.45i)T+(−51.0+29.5i)T2 1 + (1.19 + 4.45i)T + (-51.0 + 29.5i)T^{2}
67 1+(−4.98+1.33i)T+(58.0−33.5i)T2 1 + (-4.98 + 1.33i)T + (58.0 - 33.5i)T^{2}
71 1+(−3.33−0.893i)T+(61.4+35.5i)T2 1 + (-3.33 - 0.893i)T + (61.4 + 35.5i)T^{2}
73 1+(4.59+7.96i)T+(−36.5+63.2i)T2 1 + (4.59 + 7.96i)T + (-36.5 + 63.2i)T^{2}
79 1+(−4.11−15.3i)T+(−68.4+39.5i)T2 1 + (-4.11 - 15.3i)T + (-68.4 + 39.5i)T^{2}
83 1+(7.68+4.43i)T+(41.5+71.8i)T2 1 + (7.68 + 4.43i)T + (41.5 + 71.8i)T^{2}
89 1+(3.82−3.82i)T−89iT2 1 + (3.82 - 3.82i)T - 89iT^{2}
97 1+(2.50−1.44i)T+(48.5−84.0i)T2 1 + (2.50 - 1.44i)T + (48.5 - 84.0i)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

−9.939281657788570714738679869211, −9.078644204456186211486184967480, −8.110653328505855733943873759906, −7.26994120419198203119415231818, −6.89994823079451597366672895161, −5.61599815479276285095369850671, −4.72081234050658647279876714523, −3.49889430437414575251587740230, −2.30066563645654147703124470657, −0.813348659036409677275626263172, 0.956498274801420065676567544051, 2.61434020251486783066623275892, 3.19868007699459553939871316434, 4.80732441185815898868704497663, 5.70640484241127817481067847492, 6.56304713866169459781763058774, 7.59762491210547666496325618456, 8.502184692101494203003470734001, 8.809920151234542546841188991422, 10.12779592997410067295454908522

Graph of the ZZ-function along the critical line