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Properties

Label 2-1815-5.4-c1-0-58
Degree 22
Conductor 18151815
Sign 0.241−0.970i0.241 - 0.970i
Analytic cond. 14.492814.4928
Root an. cond. 3.806943.80694
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  + 1.53i·2-s − i·3-s − 0.369·4-s + (2.17 + 0.539i)5-s + 1.53·6-s + 0.290i·7-s + 2.51i·8-s − 9-s + (−0.829 + 3.34i)10-s + 0.369i·12-s − 6.97i·13-s − 0.447·14-s + (0.539 − 2.17i)15-s − 4.60·16-s + 4.78i·17-s − 1.53i·18-s + ⋯
L(s)  = 1  + 1.08i·2-s − 0.577i·3-s − 0.184·4-s + (0.970 + 0.241i)5-s + 0.628·6-s + 0.109i·7-s + 0.887i·8-s − 0.333·9-s + (−0.262 + 1.05i)10-s + 0.106i·12-s − 1.93i·13-s − 0.119·14-s + (0.139 − 0.560i)15-s − 1.15·16-s + 1.16i·17-s − 0.362i·18-s + ⋯

Functional equation

Λ(s)=(1815s/2 ΓC(s) L(s)=((0.241−0.970i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1815s/2 ΓC(s+1/2) L(s)=((0.241−0.970i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18151815    =    3⋅5⋅1123 \cdot 5 \cdot 11^{2}
Sign: 0.241−0.970i0.241 - 0.970i
Analytic conductor: 14.492814.4928
Root analytic conductor: 3.806943.80694
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1815(364,⋅)\chi_{1815} (364, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1815, ( :1/2), 0.241−0.970i)(2,\ 1815,\ (\ :1/2),\ 0.241 - 0.970i)

Particular Values

L(1)L(1) ≈\approx 2.3730602452.373060245
L(12)L(\frac12) ≈\approx 2.3730602452.373060245
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+iT 1 + iT
5 1+(−2.17−0.539i)T 1 + (-2.17 - 0.539i)T
11 1 1
good2 1−1.53iT−2T2 1 - 1.53iT - 2T^{2}
7 1−0.290iT−7T2 1 - 0.290iT - 7T^{2}
13 1+6.97iT−13T2 1 + 6.97iT - 13T^{2}
17 1−4.78iT−17T2 1 - 4.78iT - 17T^{2}
19 1−7.75T+19T2 1 - 7.75T + 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
29 1+7.41T+29T2 1 + 7.41T + 29T^{2}
31 1−6.34T+31T2 1 - 6.34T + 31T^{2}
37 1−3.41iT−37T2 1 - 3.41iT - 37T^{2}
41 1−7.41T+41T2 1 - 7.41T + 41T^{2}
43 1+0.290iT−43T2 1 + 0.290iT - 43T^{2}
47 1+5.26iT−47T2 1 + 5.26iT - 47T^{2}
53 1−5.75iT−53T2 1 - 5.75iT - 53T^{2}
59 1+3.60T+59T2 1 + 3.60T + 59T^{2}
61 1−6.68T+61T2 1 - 6.68T + 61T^{2}
67 1+6.15iT−67T2 1 + 6.15iT - 67T^{2}
71 1+5.07T+71T2 1 + 5.07T + 71T^{2}
73 1−1.12iT−73T2 1 - 1.12iT - 73T^{2}
79 1+0.921T+79T2 1 + 0.921T + 79T^{2}
83 1+1.70iT−83T2 1 + 1.70iT - 83T^{2}
89 1+4.34T+89T2 1 + 4.34T + 89T^{2}
97 1+4.68iT−97T2 1 + 4.68iT - 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

−9.266628729254300117461823112976, −8.327895060186270712077670120051, −7.66800449360478882212970532168, −7.16537220243693997009522460304, −6.03734412555635897566382552551, −5.75341689450836023137391116210, −5.12486012461702985151094102768, −3.35917160604929187125475277808, −2.48181600999973805300676977255, −1.25587853126072217047620392194, 1.01781725162231601011668709410, 2.14226703114687434567043100894, 2.89545046993389894552480190875, 4.03316793998777281659075572231, 4.75875465007929563546123876387, 5.74539741918098101320367062390, 6.71216654779773861817881812962, 7.39381261156785909510866294742, 8.883350565312083502115241051745, 9.477325992216968582096089911276

Graph of the ZZ-function along the critical line