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Properties

Label 1-1480-1480.1109-r0-0-0
Degree 11
Conductor 14801480
Sign 11
Analytic cond. 6.873096.87309
Root an. cond. 6.873096.87309
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s − 11-s − 13-s + 17-s + 19-s − 21-s + 23-s + 27-s + 29-s − 31-s − 33-s − 39-s + 41-s − 43-s − 47-s + 49-s + 51-s + 53-s + 57-s + 59-s + 61-s − 63-s + 67-s + 69-s + 71-s + ⋯
L(s)  = 1  + 3-s − 7-s + 9-s − 11-s − 13-s + 17-s + 19-s − 21-s + 23-s + 27-s + 29-s − 31-s − 33-s − 39-s + 41-s − 43-s − 47-s + 49-s + 51-s + 53-s + 57-s + 59-s + 61-s − 63-s + 67-s + 69-s + 71-s + ⋯

Functional equation

Λ(s)=(1480s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(1480s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 14801480    =    23⋅5⋅372^{3} \cdot 5 \cdot 37
Sign: 11
Analytic conductor: 6.873096.87309
Root analytic conductor: 6.873096.87309
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ1480(1109,⋅)\chi_{1480} (1109, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (1, 1480, (0: ), 1)(1,\ 1480,\ (0:\ ),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9684906561.968490656
L(12)L(\frac12) ≈\approx 1.9684906561.968490656
L(1)L(1) ≈\approx 1.3481632951.348163295
L(1)L(1) ≈\approx 1.3481632951.348163295

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
37 1 1
good3 1+T 1 + T
7 1−T 1 - T
11 1−T 1 - T
13 1−T 1 - T
17 1+T 1 + T
19 1+T 1 + T
23 1+T 1 + T
29 1+T 1 + T
31 1−T 1 - T
41 1+T 1 + T
43 1−T 1 - T
47 1−T 1 - T
53 1+T 1 + T
59 1+T 1 + T
61 1+T 1 + T
67 1+T 1 + T
71 1+T 1 + T
73 1−T 1 - T
79 1−T 1 - T
83 1+T 1 + T
89 1−T 1 - T
97 1+T 1 + T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−20.597044878934877592344022715104, −19.82432264340175996615369024496, −19.27242929318491127413325967534, −18.596992281382950001199286519007, −17.86723211403599037252990429206, −16.65127924649530894434576803646, −16.102181655726558264935460066103, −15.34766332592801934993499208118, −14.58889106351705004008504604803, −13.89046088132977626793181711089, −12.93408911964553979815099178244, −12.66899382534337587483868959446, −11.57752933663650910841670425152, −10.24005501312088998266006188816, −9.927699293484388494676159659395, −9.146022499354062083976879229467, −8.21059444698359270110404980695, −7.39452902264587061698817397145, −6.88710095229229850050824509391, −5.56335888627716199056787440663, −4.82892374697454440439774692989, −3.57419746510904027057169045618, −2.992709775420941766491946683463, −2.27205810753055573815973483819, −0.88221143063316516468833070542, 0.88221143063316516468833070542, 2.27205810753055573815973483819, 2.992709775420941766491946683463, 3.57419746510904027057169045618, 4.82892374697454440439774692989, 5.56335888627716199056787440663, 6.88710095229229850050824509391, 7.39452902264587061698817397145, 8.21059444698359270110404980695, 9.146022499354062083976879229467, 9.927699293484388494676159659395, 10.24005501312088998266006188816, 11.57752933663650910841670425152, 12.66899382534337587483868959446, 12.93408911964553979815099178244, 13.89046088132977626793181711089, 14.58889106351705004008504604803, 15.34766332592801934993499208118, 16.102181655726558264935460066103, 16.65127924649530894434576803646, 17.86723211403599037252990429206, 18.596992281382950001199286519007, 19.27242929318491127413325967534, 19.82432264340175996615369024496, 20.597044878934877592344022715104

Graph of the ZZ-function along the critical line