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Properties

Label 2-950-5.4-c1-0-4
Degree 22
Conductor 950950
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 7.585787.58578
Root an. cond. 2.754232.75423
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 + 3.03i·3-s − 4-s − 3.03·6-s + 2.46i·7-s − i·8-s − 6.19·9-s + 0.728·11-s − 3.03i·12-s + 6.23i·13-s − 2.46·14-s + 16-s + 0.563i·17-s − 6.19i·18-s + 19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.75i·3-s − 0.5·4-s − 1.23·6-s + 0.933i·7-s − 0.353i·8-s − 2.06·9-s + 0.219·11-s − 0.875i·12-s + 1.72i·13-s − 0.660·14-s + 0.250·16-s + 0.136i·17-s − 1.46i·18-s + 0.229·19-s + ⋯

Functional equation

Λ(s)=(950s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(950s/2 ΓC(s+1/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 950950    =    2⋅52⋅192 \cdot 5^{2} \cdot 19
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 7.585787.58578
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ950(799,⋅)\chi_{950} (799, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 950, ( :1/2), −0.447+0.894i)(2,\ 950,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.599132−0.969417i0.599132 - 0.969417i
L(12)L(\frac12) ≈\approx 0.599132−0.969417i0.599132 - 0.969417i
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
5 1 1
19 1−T 1 - T
good3 1−3.03iT−3T2 1 - 3.03iT - 3T^{2}
7 1−2.46iT−7T2 1 - 2.46iT - 7T^{2}
11 1−0.728T+11T2 1 - 0.728T + 11T^{2}
13 1−6.23iT−13T2 1 - 6.23iT - 13T^{2}
17 1−0.563iT−17T2 1 - 0.563iT - 17T^{2}
23 1+4.63iT−23T2 1 + 4.63iT - 23T^{2}
29 1+10.2T+29T2 1 + 10.2T + 29T^{2}
31 1−6.06T+31T2 1 - 6.06T + 31T^{2}
37 1+5.72iT−37T2 1 + 5.72iT - 37T^{2}
41 1−4.79T+41T2 1 - 4.79T + 41T^{2}
43 1+8.06iT−43T2 1 + 8.06iT - 43T^{2}
47 1−8.12iT−47T2 1 - 8.12iT - 47T^{2}
53 1−1.53iT−53T2 1 - 1.53iT - 53T^{2}
59 1−5.76T+59T2 1 - 5.76T + 59T^{2}
61 1−10.9T+61T2 1 - 10.9T + 61T^{2}
67 1−12.9iT−67T2 1 - 12.9iT - 67T^{2}
71 1+4.39T+71T2 1 + 4.39T + 71T^{2}
73 1−4.09iT−73T2 1 - 4.09iT - 73T^{2}
79 1+15.3T+79T2 1 + 15.3T + 79T^{2}
83 1+7.85iT−83T2 1 + 7.85iT - 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1−11.0iT−97T2 1 - 11.0iT - 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

−10.37772803262404080473937016563, −9.492811624209865164620765616155, −9.050089046060552111810945216321, −8.500693345445627374180009519947, −7.15168726712593702235939717049, −6.05540930528989514022884749402, −5.39537682621277548494733873925, −4.39348245550699026646732106531, −3.86774574820099920160136304989, −2.42777354679122028722842091150, 0.55640147237257796701151293681, 1.47669690046997311806703958247, 2.74498952232078345304881435295, 3.67831137126918413856069349132, 5.22960580558928601152261085071, 6.07858436103319907958096065958, 7.16663547524415812857534170425, 7.74242295226464328730948966670, 8.374406721507450405134876753779, 9.584365608553994984386940221385

Graph of the ZZ-function along the critical line