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Properties

Label 2-1805-5.4-c1-0-131
Degree 22
Conductor 18051805
Sign −0.362+0.931i-0.362 + 0.931i
Analytic cond. 14.412914.4129
Root an. cond. 3.796443.79644
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  − 0.717i·2-s + 1.56i·3-s + 1.48·4-s + (0.811 − 2.08i)5-s + 1.12·6-s − 2.51i·7-s − 2.49i·8-s + 0.535·9-s + (−1.49 − 0.581i)10-s − 5.85·11-s + 2.33i·12-s − 0.791i·13-s − 1.80·14-s + (3.27 + 1.27i)15-s + 1.17·16-s − 0.651i·17-s + ⋯
L(s)  = 1  − 0.507i·2-s + 0.906i·3-s + 0.742·4-s + (0.362 − 0.931i)5-s + 0.459·6-s − 0.952i·7-s − 0.883i·8-s + 0.178·9-s + (−0.472 − 0.183i)10-s − 1.76·11-s + 0.673i·12-s − 0.219i·13-s − 0.482·14-s + (0.844 + 0.328i)15-s + 0.294·16-s − 0.157i·17-s + ⋯

Functional equation

Λ(s)=(1805s/2 ΓC(s) L(s)=((−0.362+0.931i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1805s/2 ΓC(s+1/2) L(s)=((−0.362+0.931i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18051805    =    5⋅1925 \cdot 19^{2}
Sign: −0.362+0.931i-0.362 + 0.931i
Analytic conductor: 14.412914.4129
Root analytic conductor: 3.796443.79644
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1805(1084,⋅)\chi_{1805} (1084, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1805, ( :1/2), −0.362+0.931i)(2,\ 1805,\ (\ :1/2),\ -0.362 + 0.931i)

Particular Values

L(1)L(1) ≈\approx 1.8522910281.852291028
L(12)L(\frac12) ≈\approx 1.8522910281.852291028
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+(−0.811+2.08i)T 1 + (-0.811 + 2.08i)T
19 1 1
good2 1+0.717iT−2T2 1 + 0.717iT - 2T^{2}
3 1−1.56iT−3T2 1 - 1.56iT - 3T^{2}
7 1+2.51iT−7T2 1 + 2.51iT - 7T^{2}
11 1+5.85T+11T2 1 + 5.85T + 11T^{2}
13 1+0.791iT−13T2 1 + 0.791iT - 13T^{2}
17 1+0.651iT−17T2 1 + 0.651iT - 17T^{2}
23 1+4.88iT−23T2 1 + 4.88iT - 23T^{2}
29 1+4.83T+29T2 1 + 4.83T + 29T^{2}
31 1−6.73T+31T2 1 - 6.73T + 31T^{2}
37 1+0.741iT−37T2 1 + 0.741iT - 37T^{2}
41 1+8.04T+41T2 1 + 8.04T + 41T^{2}
43 1+0.761iT−43T2 1 + 0.761iT - 43T^{2}
47 1+11.3iT−47T2 1 + 11.3iT - 47T^{2}
53 1−12.8iT−53T2 1 - 12.8iT - 53T^{2}
59 1+2.14T+59T2 1 + 2.14T + 59T^{2}
61 1+6.75T+61T2 1 + 6.75T + 61T^{2}
67 1+13.8iT−67T2 1 + 13.8iT - 67T^{2}
71 1−6.05T+71T2 1 - 6.05T + 71T^{2}
73 1−11.1iT−73T2 1 - 11.1iT - 73T^{2}
79 1−15.7T+79T2 1 - 15.7T + 79T^{2}
83 1−3.26iT−83T2 1 - 3.26iT - 83T^{2}
89 1−1.07T+89T2 1 - 1.07T + 89T^{2}
97 1+10.1iT−97T2 1 + 10.1iT - 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

−9.326382454071176323499536048024, −8.236660779519274593713517803219, −7.56622029258066328831517412631, −6.66504696758620456812709859157, −5.53411556051841310633161385357, −4.81773103671980383158846856585, −4.04271204301533250823181228529, −3.03289984542557293077449402670, −1.96209350994045207308065939452, −0.62066247085532964721675995595, 1.80196201287786673139558657242, 2.42116606800860355961030921236, 3.21326508617692645954131783083, 5.03830275387404548409092611783, 5.79241341846132851285305440385, 6.38513792485628329890606601265, 7.15895274198292110982864868846, 7.76133916780733040088729980106, 8.282314249552113211680154836711, 9.568968594564731856827669319764

Graph of the ZZ-function along the critical line