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Properties

Label 2-845-65.18-c1-0-62
Degree 22
Conductor 845845
Sign −0.638−0.769i-0.638 - 0.769i
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  − 1.58i·2-s + (−0.139 − 0.139i)3-s − 0.511·4-s + (−2.23 − 0.0672i)5-s + (−0.221 + 0.221i)6-s − 0.548·7-s − 2.35i·8-s − 2.96i·9-s + (−0.106 + 3.54i)10-s + (−0.108 − 0.108i)11-s + (0.0713 + 0.0713i)12-s + 0.868i·14-s + (0.302 + 0.321i)15-s − 4.76·16-s + (2.22 + 2.22i)17-s − 4.69·18-s + ⋯
L(s)  = 1  − 1.12i·2-s + (−0.0805 − 0.0805i)3-s − 0.255·4-s + (−0.999 − 0.0300i)5-s + (−0.0902 + 0.0902i)6-s − 0.207·7-s − 0.834i·8-s − 0.987i·9-s + (−0.0337 + 1.12i)10-s + (−0.0326 − 0.0326i)11-s + (0.0205 + 0.0205i)12-s + 0.232i·14-s + (0.0780 + 0.0829i)15-s − 1.19·16-s + (0.539 + 0.539i)17-s − 1.10·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((−0.638−0.769i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((−0.638−0.769i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: −0.638−0.769i-0.638 - 0.769i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(408,⋅)\chi_{845} (408, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), −0.638−0.769i)(2,\ 845,\ (\ :1/2),\ -0.638 - 0.769i)

Particular Values

L(1)L(1) ≈\approx 0.258375+0.550492i0.258375 + 0.550492i
L(12)L(\frac12) ≈\approx 0.258375+0.550492i0.258375 + 0.550492i
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+(2.23+0.0672i)T 1 + (2.23 + 0.0672i)T
13 1 1
good2 1+1.58iT−2T2 1 + 1.58iT - 2T^{2}
3 1+(0.139+0.139i)T+3iT2 1 + (0.139 + 0.139i)T + 3iT^{2}
7 1+0.548T+7T2 1 + 0.548T + 7T^{2}
11 1+(0.108+0.108i)T+11iT2 1 + (0.108 + 0.108i)T + 11iT^{2}
17 1+(−2.22−2.22i)T+17iT2 1 + (-2.22 - 2.22i)T + 17iT^{2}
19 1+(3.22+3.22i)T+19iT2 1 + (3.22 + 3.22i)T + 19iT^{2}
23 1+(2.50−2.50i)T−23iT2 1 + (2.50 - 2.50i)T - 23iT^{2}
29 1+2.34iT−29T2 1 + 2.34iT - 29T^{2}
31 1+(6.60−6.60i)T−31iT2 1 + (6.60 - 6.60i)T - 31iT^{2}
37 1+6.80T+37T2 1 + 6.80T + 37T^{2}
41 1+(−2.53+2.53i)T−41iT2 1 + (-2.53 + 2.53i)T - 41iT^{2}
43 1+(5.02−5.02i)T−43iT2 1 + (5.02 - 5.02i)T - 43iT^{2}
47 1+9.13T+47T2 1 + 9.13T + 47T^{2}
53 1+(3.70+3.70i)T+53iT2 1 + (3.70 + 3.70i)T + 53iT^{2}
59 1+(−2.69+2.69i)T−59iT2 1 + (-2.69 + 2.69i)T - 59iT^{2}
61 1−7.84T+61T2 1 - 7.84T + 61T^{2}
67 1+4.89iT−67T2 1 + 4.89iT - 67T^{2}
71 1+(−11.0+11.0i)T−71iT2 1 + (-11.0 + 11.0i)T - 71iT^{2}
73 1−3.91iT−73T2 1 - 3.91iT - 73T^{2}
79 1+11.1iT−79T2 1 + 11.1iT - 79T^{2}
83 1−13.4T+83T2 1 - 13.4T + 83T^{2}
89 1+(−6.43+6.43i)T−89iT2 1 + (-6.43 + 6.43i)T - 89iT^{2}
97 1−7.57iT−97T2 1 - 7.57iT - 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

−9.813531517488949833773241159383, −9.022384412816950880108084249566, −8.073926548400572392854488028879, −6.98949670456398765665117019247, −6.35498362077324199165232437139, −4.86726397109504855834782859789, −3.64698611538687796641773578112, −3.32107051303554426607471307149, −1.73908522101519657724997704677, −0.28607700011836006863388936583, 2.18523734740445213715844425393, 3.62104387301692896998334853136, 4.75848074565500040057226147084, 5.52699459590328034461269974809, 6.57956571689878017282733165414, 7.36266807977105522439502174332, 8.047759483821567392365443218194, 8.562644107768739774515673174767, 9.852027598829205780719231582183, 10.82799506168554732527381515586

Graph of the ZZ-function along the critical line