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Properties

Label 2-845-65.32-c1-0-47
Degree 22
Conductor 845845
Sign −0.151+0.988i-0.151 + 0.988i
Analytic cond. 6.747356.74735
Root an. cond. 2.597562.59756
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  + (1.32 − 2.29i)2-s + (0.335 + 1.25i)3-s + (−2.51 − 4.34i)4-s + (1.30 + 1.81i)5-s + (3.31 + 0.889i)6-s + (0.0972 − 0.0561i)7-s − 8.00·8-s + (1.14 − 0.658i)9-s + (5.89 − 0.585i)10-s + (1.78 − 0.479i)11-s + (4.60 − 4.60i)12-s − 0.297i·14-s + (−1.83 + 2.24i)15-s + (−5.58 + 9.67i)16-s + (2.63 + 0.706i)17-s − 3.49i·18-s + ⋯
L(s)  = 1  + (0.936 − 1.62i)2-s + (0.193 + 0.723i)3-s + (−1.25 − 2.17i)4-s + (0.583 + 0.812i)5-s + (1.35 + 0.363i)6-s + (0.0367 − 0.0212i)7-s − 2.83·8-s + (0.380 − 0.219i)9-s + (1.86 − 0.185i)10-s + (0.539 − 0.144i)11-s + (1.32 − 1.32i)12-s − 0.0795i·14-s + (−0.474 + 0.579i)15-s + (−1.39 + 2.41i)16-s + (0.639 + 0.171i)17-s − 0.823i·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((−0.151+0.988i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((−0.151+0.988i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: −0.151+0.988i-0.151 + 0.988i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(357,⋅)\chi_{845} (357, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), −0.151+0.988i)(2,\ 845,\ (\ :1/2),\ -0.151 + 0.988i)

Particular Values

L(1)L(1) ≈\approx 1.89033−2.20246i1.89033 - 2.20246i
L(12)L(\frac12) ≈\approx 1.89033−2.20246i1.89033 - 2.20246i
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)
bad5 1+(−1.30−1.81i)T 1 + (-1.30 - 1.81i)T
13 1 1
good2 1+(−1.32+2.29i)T+(−1−1.73i)T2 1 + (-1.32 + 2.29i)T + (-1 - 1.73i)T^{2}
3 1+(−0.335−1.25i)T+(−2.59+1.5i)T2 1 + (-0.335 - 1.25i)T + (-2.59 + 1.5i)T^{2}
7 1+(−0.0972+0.0561i)T+(3.5−6.06i)T2 1 + (-0.0972 + 0.0561i)T + (3.5 - 6.06i)T^{2}
11 1+(−1.78+0.479i)T+(9.52−5.5i)T2 1 + (-1.78 + 0.479i)T + (9.52 - 5.5i)T^{2}
17 1+(−2.63−0.706i)T+(14.7+8.5i)T2 1 + (-2.63 - 0.706i)T + (14.7 + 8.5i)T^{2}
19 1+(−1.80+6.72i)T+(−16.4−9.5i)T2 1 + (-1.80 + 6.72i)T + (-16.4 - 9.5i)T^{2}
23 1+(−3.10+0.831i)T+(19.9−11.5i)T2 1 + (-3.10 + 0.831i)T + (19.9 - 11.5i)T^{2}
29 1+(4.03+2.32i)T+(14.5+25.1i)T2 1 + (4.03 + 2.32i)T + (14.5 + 25.1i)T^{2}
31 1+(−0.624+0.624i)T−31iT2 1 + (-0.624 + 0.624i)T - 31iT^{2}
37 1+(−1.27−0.737i)T+(18.5+32.0i)T2 1 + (-1.27 - 0.737i)T + (18.5 + 32.0i)T^{2}
41 1+(1.40+5.24i)T+(−35.5+20.5i)T2 1 + (1.40 + 5.24i)T + (-35.5 + 20.5i)T^{2}
43 1+(1.00−3.76i)T+(−37.2−21.5i)T2 1 + (1.00 - 3.76i)T + (-37.2 - 21.5i)T^{2}
47 1−0.345iT−47T2 1 - 0.345iT - 47T^{2}
53 1+(3.59−3.59i)T−53iT2 1 + (3.59 - 3.59i)T - 53iT^{2}
59 1+(1.24+0.332i)T+(51.0+29.5i)T2 1 + (1.24 + 0.332i)T + (51.0 + 29.5i)T^{2}
61 1+(−1.39−2.41i)T+(−30.5+52.8i)T2 1 + (-1.39 - 2.41i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.0721−0.124i)T+(−33.5−58.0i)T2 1 + (0.0721 - 0.124i)T + (-33.5 - 58.0i)T^{2}
71 1+(5.28+1.41i)T+(61.4+35.5i)T2 1 + (5.28 + 1.41i)T + (61.4 + 35.5i)T^{2}
73 1+9.06T+73T2 1 + 9.06T + 73T^{2}
79 1−15.1iT−79T2 1 - 15.1iT - 79T^{2}
83 1−8.53iT−83T2 1 - 8.53iT - 83T^{2}
89 1+(0.147+0.549i)T+(−77.0+44.5i)T2 1 + (0.147 + 0.549i)T + (-77.0 + 44.5i)T^{2}
97 1+(7.48+12.9i)T+(−48.5+84.0i)T2 1 + (7.48 + 12.9i)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

−10.08131896223420051157823999506, −9.569718718599766744194494391138, −8.962592964488677947014836921421, −7.15292242865437169307860493769, −6.10797936985328110828503940873, −5.13571069131743323777397297444, −4.25794536227236698985234775791, −3.36141033169689710502551267897, −2.65152251112209191334173104521, −1.29153401544000360916509654711, 1.58588763845296692621616874732, 3.39867823083885514226506457164, 4.49131992088152941422634189942, 5.32947686957878927261186393757, 6.06191947662110188752811191671, 6.92213396767192548054717497296, 7.69708504806201772758498255182, 8.308098274931752324338297622556, 9.199366906562872408760308068298, 10.09539361837613217553247393789

Graph of the ZZ-function along the critical line