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Properties

Label 2-390-39.8-c1-0-7
Degree 22
Conductor 390390
Sign −0.738+0.674i-0.738 + 0.674i
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  + (−0.707 + 0.707i)2-s + (−1.58 − 0.707i)3-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + (1.61 − 0.618i)6-s + (0.707 + 0.707i)8-s + (2.00 + 2.23i)9-s − 1.00i·10-s + (1.41 + 1.41i)11-s + (−0.707 + 1.58i)12-s + (0.418 − 3.58i)13-s + (1.61 − 0.618i)15-s − 1.00·16-s − 7.30·17-s + (−2.99 − 0.166i)18-s + (−5.16 − 5.16i)19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.912 − 0.408i)3-s − 0.500i·4-s + (−0.316 + 0.316i)5-s + (0.660 − 0.252i)6-s + (0.250 + 0.250i)8-s + (0.666 + 0.745i)9-s − 0.316i·10-s + (0.426 + 0.426i)11-s + (−0.204 + 0.456i)12-s + (0.116 − 0.993i)13-s + (0.417 − 0.159i)15-s − 0.250·16-s − 1.77·17-s + (−0.706 − 0.0393i)18-s + (−1.18 − 1.18i)19-s + ⋯

Functional equation

Λ(s)=(390s/2 ΓC(s) L(s)=((−0.738+0.674i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(390s/2 ΓC(s+1/2) L(s)=((−0.738+0.674i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 390390    =    2⋅3⋅5⋅132 \cdot 3 \cdot 5 \cdot 13
Sign: −0.738+0.674i-0.738 + 0.674i
Analytic conductor: 3.114163.11416
Root analytic conductor: 1.764691.76469
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ390(281,⋅)\chi_{390} (281, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 390, ( :1/2), −0.738+0.674i)(2,\ 390,\ (\ :1/2),\ -0.738 + 0.674i)

Particular Values

L(1)L(1) ≈\approx 0.0571069−0.147272i0.0571069 - 0.147272i
L(12)L(\frac12) ≈\approx 0.0571069−0.147272i0.0571069 - 0.147272i
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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
3 1+(1.58+0.707i)T 1 + (1.58 + 0.707i)T
5 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
13 1+(−0.418+3.58i)T 1 + (-0.418 + 3.58i)T
good7 1−7iT2 1 - 7iT^{2}
11 1+(−1.41−1.41i)T+11iT2 1 + (-1.41 - 1.41i)T + 11iT^{2}
17 1+7.30T+17T2 1 + 7.30T + 17T^{2}
19 1+(5.16+5.16i)T+19iT2 1 + (5.16 + 5.16i)T + 19iT^{2}
23 1+4.47T+23T2 1 + 4.47T + 23T^{2}
29 1−4.47iT−29T2 1 - 4.47iT - 29T^{2}
31 1+(3+3i)T+31iT2 1 + (3 + 3i)T + 31iT^{2}
37 1+(4−4i)T−37iT2 1 + (4 - 4i)T - 37iT^{2}
41 1+(−2.23+2.23i)T−41iT2 1 + (-2.23 + 2.23i)T - 41iT^{2}
43 1+7.16iT−43T2 1 + 7.16iT - 43T^{2}
47 1+(7.30+7.30i)T+47iT2 1 + (7.30 + 7.30i)T + 47iT^{2}
53 1+1.41iT−53T2 1 + 1.41iT - 53T^{2}
59 1+(5.88+5.88i)T+59iT2 1 + (5.88 + 5.88i)T + 59iT^{2}
61 1−10.6T+61T2 1 - 10.6T + 61T^{2}
67 1+(−7.16−7.16i)T+67iT2 1 + (-7.16 - 7.16i)T + 67iT^{2}
71 1+(−3.87+3.87i)T−71iT2 1 + (-3.87 + 3.87i)T - 71iT^{2}
73 1+(5.16−5.16i)T−73iT2 1 + (5.16 - 5.16i)T - 73iT^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1+(−5.65+5.65i)T−83iT2 1 + (-5.65 + 5.65i)T - 83iT^{2}
89 1+(0.592+0.592i)T+89iT2 1 + (0.592 + 0.592i)T + 89iT^{2}
97 1+(1.16+1.16i)T+97iT2 1 + (1.16 + 1.16i)T + 97iT^{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

−10.90191209445210660100492773787, −10.27031139681549510500757589569, −8.998514229484911037507261386764, −8.074995037883201708109391584395, −6.93361180419447513672259264674, −6.53702416903514063514953488713, −5.29935748258280637832674056373, −4.22620033361907259745018017533, −2.11985201196596893417599245391, −0.13265617460980325265062695341, 1.81314899718030458089601765233, 3.88715925731325827312995461161, 4.48641444094425686499075859946, 6.05820037415172688065487419600, 6.81269338436042885439814493699, 8.237116444239579725139441951724, 9.059668914600270527530384009008, 9.915981090677610777242798700949, 10.96491905749480010028692554956, 11.41095687713238982829668626139

Graph of the ZZ-function along the critical line