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Properties

Label 2-1014-13.12-c1-0-25
Degree 22
Conductor 10141014
Sign 0.0304+0.999i0.0304 + 0.999i
Analytic cond. 8.096838.09683
Root an. cond. 2.845492.84549
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 − 4.04i·5-s + i·6-s + 0.692i·7-s − i·8-s + 9-s + 4.04·10-s − 4.85i·11-s − 12-s − 0.692·14-s − 4.04i·15-s + 16-s − 7.38·17-s + i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 1.81i·5-s + 0.408i·6-s + 0.261i·7-s − 0.353i·8-s + 0.333·9-s + 1.28·10-s − 1.46i·11-s − 0.288·12-s − 0.184·14-s − 1.04i·15-s + 0.250·16-s − 1.79·17-s + 0.235i·18-s + ⋯

Functional equation

Λ(s)=(1014s/2 ΓC(s) L(s)=((0.0304+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1014s/2 ΓC(s+1/2) L(s)=((0.0304+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10141014    =    2⋅3⋅1322 \cdot 3 \cdot 13^{2}
Sign: 0.0304+0.999i0.0304 + 0.999i
Analytic conductor: 8.096838.09683
Root analytic conductor: 2.845492.84549
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1014(337,⋅)\chi_{1014} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1014, ( :1/2), 0.0304+0.999i)(2,\ 1014,\ (\ :1/2),\ 0.0304 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.3116807981.311680798
L(12)L(\frac12) ≈\approx 1.3116807981.311680798
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
13 1 1
good5 1+4.04iT−5T2 1 + 4.04iT - 5T^{2}
7 1−0.692iT−7T2 1 - 0.692iT - 7T^{2}
11 1+4.85iT−11T2 1 + 4.85iT - 11T^{2}
17 1+7.38T+17T2 1 + 7.38T + 17T^{2}
19 1−1.78iT−19T2 1 - 1.78iT - 19T^{2}
23 1+5.10T+23T2 1 + 5.10T + 23T^{2}
29 1+3.34T+29T2 1 + 3.34T + 29T^{2}
31 1+0.972iT−31T2 1 + 0.972iT - 31T^{2}
37 1−1.28iT−37T2 1 - 1.28iT - 37T^{2}
41 1+1.50iT−41T2 1 + 1.50iT - 41T^{2}
43 1−8.31T+43T2 1 - 8.31T + 43T^{2}
47 1+7.20iT−47T2 1 + 7.20iT - 47T^{2}
53 1−13.4T+53T2 1 - 13.4T + 53T^{2}
59 1−1.30iT−59T2 1 - 1.30iT - 59T^{2}
61 1+0.396T+61T2 1 + 0.396T + 61T^{2}
67 1+6.05iT−67T2 1 + 6.05iT - 67T^{2}
71 1−1.32iT−71T2 1 - 1.32iT - 71T^{2}
73 1+7.65iT−73T2 1 + 7.65iT - 73T^{2}
79 1+8.33T+79T2 1 + 8.33T + 79T^{2}
83 1+15.3iT−83T2 1 + 15.3iT - 83T^{2}
89 1−3.10iT−89T2 1 - 3.10iT - 89T^{2}
97 1−8.54iT−97T2 1 - 8.54iT - 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

−9.197688213805736506823808132532, −8.798267522111578953997592293996, −8.365398230448837643681388581185, −7.47198525089832125550909266730, −6.13491554949825851366793948114, −5.53731470480394207642060446762, −4.48838879773782674869707215619, −3.77955172827098274832144802424, −2.06510733659147620384261976194, −0.53068843562592457290458618166, 2.13397469367192747351334117366, 2.54203323910779082045144166718, 3.82253387853839169425188470607, 4.44576254840836318438154960731, 6.05075647708314483753185269432, 7.10649551024888338897072314165, 7.38757951516034581991644915916, 8.666359133629152854343203103407, 9.624030967868472264902246746072, 10.18309634630888290225387004116

Graph of the ZZ-function along the critical line