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Properties

Label 2-1575-5.4-c1-0-44
Degree 22
Conductor 15751575
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 12.576412.5764
Root an. cond. 3.546323.54632
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  − 1.56i·2-s − 0.438·4-s − i·7-s − 2.43i·8-s − 2.56·11-s − 4.56i·13-s − 1.56·14-s − 4.68·16-s + 4.56i·17-s − 1.12·19-s + 4i·22-s − 5.12i·23-s − 7.12·26-s + 0.438i·28-s − 5.68·29-s + ⋯
L(s)  = 1  − 1.10i·2-s − 0.219·4-s − 0.377i·7-s − 0.862i·8-s − 0.772·11-s − 1.26i·13-s − 0.417·14-s − 1.17·16-s + 1.10i·17-s − 0.257·19-s + 0.852i·22-s − 1.06i·23-s − 1.39·26-s + 0.0828i·28-s − 1.05·29-s + ⋯

Functional equation

Λ(s)=(1575s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1575s/2 ΓC(s+1/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15751575    =    32⋅52⋅73^{2} \cdot 5^{2} \cdot 7
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 12.576412.5764
Root analytic conductor: 3.546323.54632
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1575(1324,⋅)\chi_{1575} (1324, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1575, ( :1/2), −0.894−0.447i)(2,\ 1575,\ (\ :1/2),\ -0.894 - 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.0900604871.090060487
L(12)L(\frac12) ≈\approx 1.0900604871.090060487
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)
bad3 1 1
5 1 1
7 1+iT 1 + iT
good2 1+1.56iT−2T2 1 + 1.56iT - 2T^{2}
11 1+2.56T+11T2 1 + 2.56T + 11T^{2}
13 1+4.56iT−13T2 1 + 4.56iT - 13T^{2}
17 1−4.56iT−17T2 1 - 4.56iT - 17T^{2}
19 1+1.12T+19T2 1 + 1.12T + 19T^{2}
23 1+5.12iT−23T2 1 + 5.12iT - 23T^{2}
29 1+5.68T+29T2 1 + 5.68T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−6iT−37T2 1 - 6iT - 37T^{2}
41 1−3.12T+41T2 1 - 3.12T + 41T^{2}
43 1+9.12iT−43T2 1 + 9.12iT - 43T^{2}
47 1+3.68iT−47T2 1 + 3.68iT - 47T^{2}
53 1−3.12iT−53T2 1 - 3.12iT - 53T^{2}
59 1+4T+59T2 1 + 4T + 59T^{2}
61 1+9.36T+61T2 1 + 9.36T + 61T^{2}
67 1+6.24iT−67T2 1 + 6.24iT - 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+4.24iT−73T2 1 + 4.24iT - 73T^{2}
79 1−6.56T+79T2 1 - 6.56T + 79T^{2}
83 1−4iT−83T2 1 - 4iT - 83T^{2}
89 1−7.12T+89T2 1 - 7.12T + 89T^{2}
97 1+14.8iT−97T2 1 + 14.8iT - 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

−9.153429493675716041371549902558, −8.207018861660650334661164819729, −7.51269387809278510023704412646, −6.50401882888075287433743111109, −5.62999699156149640233807671369, −4.51626160158364359327914286232, −3.58267958984426480832226036404, −2.77709823792998055210940396600, −1.74880183530616904935616320243, −0.39029540564452322454680269083, 1.88905477895534508055376154293, 2.88872793543652654702833825380, 4.31392834042464665713594538837, 5.20199900037877908056199164458, 5.86960166572514984181997014938, 6.72996637630738448129875897163, 7.47447093172145895430270401050, 7.985896817667059940479870803953, 9.157856435115486013536635343578, 9.388513772145679746426722837876

Graph of the ZZ-function along the critical line