Properties

Label 2-1274-7.2-c1-0-34
Degree 22
Conductor 12741274
Sign 0.9910.126i-0.991 - 0.126i
Analytic cond. 10.172910.1729
Root an. cond. 3.189503.18950
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  + (−0.5 + 0.866i)2-s + (−1.5 − 2.59i)3-s + (−0.499 − 0.866i)4-s + (2 − 3.46i)5-s + 3·6-s + 0.999·8-s + (−3 + 5.19i)9-s + (1.99 + 3.46i)10-s + (−0.5 − 0.866i)11-s + (−1.50 + 2.59i)12-s − 13-s − 12·15-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)18-s + (3 − 5.19i)19-s − 3.99·20-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.866 − 1.49i)3-s + (−0.249 − 0.433i)4-s + (0.894 − 1.54i)5-s + 1.22·6-s + 0.353·8-s + (−1 + 1.73i)9-s + (0.632 + 1.09i)10-s + (−0.150 − 0.261i)11-s + (−0.433 + 0.749i)12-s − 0.277·13-s − 3.09·15-s + (−0.125 + 0.216i)16-s + (−0.707 − 1.22i)18-s + (0.688 − 1.19i)19-s − 0.894·20-s + ⋯

Functional equation

Λ(s)=(1274s/2ΓC(s)L(s)=((0.9910.126i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1274s/2ΓC(s+1/2)L(s)=((0.9910.126i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12741274    =    272132 \cdot 7^{2} \cdot 13
Sign: 0.9910.126i-0.991 - 0.126i
Analytic conductor: 10.172910.1729
Root analytic conductor: 3.189503.18950
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1274(79,)\chi_{1274} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1274, ( :1/2), 0.9910.126i)(2,\ 1274,\ (\ :1/2),\ -0.991 - 0.126i)

Particular Values

L(1)L(1) \approx 0.80709263670.8070926367
L(12)L(\frac12) \approx 0.80709263670.8070926367
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
7 1 1
13 1+T 1 + T
good3 1+(1.5+2.59i)T+(1.5+2.59i)T2 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2}
5 1+(2+3.46i)T+(2.54.33i)T2 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2}
11 1+(0.5+0.866i)T+(5.5+9.52i)T2 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2}
17 1+(8.5+14.7i)T2 1 + (-8.5 + 14.7i)T^{2}
19 1+(3+5.19i)T+(9.516.4i)T2 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2}
23 1+(3.5+6.06i)T+(11.519.9i)T2 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2}
29 1+4T+29T2 1 + 4T + 29T^{2}
31 1+(3.5+6.06i)T+(15.5+26.8i)T2 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2}
37 1+(4.57.79i)T+(18.532.0i)T2 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2}
41 1+3T+41T2 1 + 3T + 41T^{2}
43 14T+43T2 1 - 4T + 43T^{2}
47 1+(3.56.06i)T+(23.540.7i)T2 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2}
53 1+(26.5+45.8i)T2 1 + (-26.5 + 45.8i)T^{2}
59 1+(58.66i)T+(29.5+51.0i)T2 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.50.866i)T+(30.552.8i)T2 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.5+0.866i)T+(33.5+58.0i)T2 1 + (0.5 + 0.866i)T + (-33.5 + 58.0i)T^{2}
71 116T+71T2 1 - 16T + 71T^{2}
73 1+(2.5+4.33i)T+(36.5+63.2i)T2 1 + (2.5 + 4.33i)T + (-36.5 + 63.2i)T^{2}
79 1+(5.59.52i)T+(39.568.4i)T2 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(3+5.19i)T+(44.577.0i)T2 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2}
97 1+T+97T2 1 + T + 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.021594127433278379638206563697, −8.382568948786264269487885069334, −7.52797768626457675565164781243, −6.73565909885860207790899875734, −5.96928570800475926006589990578, −5.28574528877152935914182696175, −4.71016423673237050421320595122, −2.37640809178145175169787267256, −1.28424668194566112350330942710, −0.45543412837843330237426882154, 1.92614253494612465849687376707, 3.30293151351504200317268486616, 3.70039284966342558337052914706, 5.21295223087484650788380698945, 5.62392069734430319538721511653, 6.74344230799443447443386479398, 7.55132031901874429827242824087, 9.051353354120912711582417601800, 9.691263019391763073838050297185, 10.13371298581512180636247451432

Graph of the ZZ-function along the critical line