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Properties

Label 2-390-13.12-c1-0-5
Degree 22
Conductor 390390
Sign 0.987−0.155i0.987 - 0.155i
Analytic cond. 3.114163.11416
Root an. cond. 1.764691.76469
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  − i·2-s + 3-s − 4-s + i·5-s − i·6-s + 3.12i·7-s + i·8-s + 9-s + 10-s + 5.12i·11-s − 12-s + (3.56 − 0.561i)13-s + 3.12·14-s + i·15-s + 16-s + 2·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s + 1.18i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s + 1.54i·11-s − 0.288·12-s + (0.987 − 0.155i)13-s + 0.834·14-s + 0.258i·15-s + 0.250·16-s + 0.485·17-s + ⋯

Functional equation

Λ(s)=(390s/2 ΓC(s) L(s)=((0.987−0.155i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(390s/2 ΓC(s+1/2) L(s)=((0.987−0.155i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 390390    =    2⋅3⋅5⋅132 \cdot 3 \cdot 5 \cdot 13
Sign: 0.987−0.155i0.987 - 0.155i
Analytic conductor: 3.114163.11416
Root analytic conductor: 1.764691.76469
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ390(181,⋅)\chi_{390} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 390, ( :1/2), 0.987−0.155i)(2,\ 390,\ (\ :1/2),\ 0.987 - 0.155i)

Particular Values

L(1)L(1) ≈\approx 1.56736+0.122805i1.56736 + 0.122805i
L(12)L(\frac12) ≈\approx 1.56736+0.122805i1.56736 + 0.122805i
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+iT 1 + iT
3 1−T 1 - T
5 1−iT 1 - iT
13 1+(−3.56+0.561i)T 1 + (-3.56 + 0.561i)T
good7 1−3.12iT−7T2 1 - 3.12iT - 7T^{2}
11 1−5.12iT−11T2 1 - 5.12iT - 11T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+6iT−19T2 1 + 6iT - 19T^{2}
23 1+5.12T+23T2 1 + 5.12T + 23T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1−3.12iT−31T2 1 - 3.12iT - 31T^{2}
37 1+5.12iT−37T2 1 + 5.12iT - 37T^{2}
41 1−0.876iT−41T2 1 - 0.876iT - 41T^{2}
43 1−6.24T+43T2 1 - 6.24T + 43T^{2}
47 1+6.24iT−47T2 1 + 6.24iT - 47T^{2}
53 1+13.3T+53T2 1 + 13.3T + 53T^{2}
59 1+1.12iT−59T2 1 + 1.12iT - 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1−4.87iT−67T2 1 - 4.87iT - 67T^{2}
71 1−10.2iT−71T2 1 - 10.2iT - 71T^{2}
73 1+13.1iT−73T2 1 + 13.1iT - 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+6.24iT−83T2 1 + 6.24iT - 83T^{2}
89 1−3.12iT−89T2 1 - 3.12iT - 89T^{2}
97 1+13.1iT−97T2 1 + 13.1iT - 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

−11.38127746873682666631590746499, −10.37101883808793684106382668017, −9.519613973867561841431976654146, −8.812024689168253307926174133572, −7.80969928564689767813418066144, −6.64359584280747701279743440924, −5.36126612194867428113126943173, −4.14944062459346395365861817459, −2.87150561423351129406409523583, −1.94216972204415676821100824926, 1.12154252455089545342960236453, 3.47828406412126673822702117232, 4.17139879793489547440662405322, 5.70088274447625252525906654435, 6.49499788494832839476608606612, 8.022360486362146270906906186783, 8.072100298636340252821930282454, 9.297585462794563626732895733261, 10.24661716179834426251987970577, 11.13981131024825043284159099711

Graph of the ZZ-function along the critical line