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Properties

Label 2-44-44.7-c1-0-2
Degree 22
Conductor 4444
Sign 0.708+0.705i0.708 + 0.705i
Analytic cond. 0.3513410.351341
Root an. cond. 0.5927400.592740
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  + (−0.0737 − 1.41i)2-s + (1.70 + 0.554i)3-s + (−1.98 + 0.208i)4-s + (−2.39 + 1.74i)5-s + (0.656 − 2.44i)6-s + (−0.815 − 2.51i)7-s + (0.440 + 2.79i)8-s + (0.174 + 0.126i)9-s + (2.63 + 3.26i)10-s + (1.40 + 3.00i)11-s + (−3.50 − 0.747i)12-s + (1.39 − 1.92i)13-s + (−3.48 + 1.33i)14-s + (−5.05 + 1.64i)15-s + (3.91 − 0.828i)16-s + (0.468 + 0.644i)17-s + ⋯
L(s)  = 1  + (−0.0521 − 0.998i)2-s + (0.984 + 0.319i)3-s + (−0.994 + 0.104i)4-s + (−1.07 + 0.779i)5-s + (0.268 − 0.999i)6-s + (−0.308 − 0.948i)7-s + (0.155 + 0.987i)8-s + (0.0580 + 0.0421i)9-s + (0.834 + 1.03i)10-s + (0.425 + 0.905i)11-s + (−1.01 − 0.215i)12-s + (0.388 − 0.534i)13-s + (−0.931 + 0.357i)14-s + (−1.30 + 0.424i)15-s + (0.978 − 0.207i)16-s + (0.113 + 0.156i)17-s + ⋯

Functional equation

Λ(s)=(44s/2 ΓC(s) L(s)=((0.708+0.705i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(44s/2 ΓC(s+1/2) L(s)=((0.708+0.705i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4444    =    22⋅112^{2} \cdot 11
Sign: 0.708+0.705i0.708 + 0.705i
Analytic conductor: 0.3513410.351341
Root analytic conductor: 0.5927400.592740
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ44(7,⋅)\chi_{44} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 44, ( :1/2), 0.708+0.705i)(2,\ 44,\ (\ :1/2),\ 0.708 + 0.705i)

Particular Values

L(1)L(1) ≈\approx 0.749457−0.309318i0.749457 - 0.309318i
L(12)L(\frac12) ≈\approx 0.749457−0.309318i0.749457 - 0.309318i
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+(0.0737+1.41i)T 1 + (0.0737 + 1.41i)T
11 1+(−1.40−3.00i)T 1 + (-1.40 - 3.00i)T
good3 1+(−1.70−0.554i)T+(2.42+1.76i)T2 1 + (-1.70 - 0.554i)T + (2.42 + 1.76i)T^{2}
5 1+(2.39−1.74i)T+(1.54−4.75i)T2 1 + (2.39 - 1.74i)T + (1.54 - 4.75i)T^{2}
7 1+(0.815+2.51i)T+(−5.66+4.11i)T2 1 + (0.815 + 2.51i)T + (-5.66 + 4.11i)T^{2}
13 1+(−1.39+1.92i)T+(−4.01−12.3i)T2 1 + (-1.39 + 1.92i)T + (-4.01 - 12.3i)T^{2}
17 1+(−0.468−0.644i)T+(−5.25+16.1i)T2 1 + (-0.468 - 0.644i)T + (-5.25 + 16.1i)T^{2}
19 1+(−0.624+1.92i)T+(−15.3−11.1i)T2 1 + (-0.624 + 1.92i)T + (-15.3 - 11.1i)T^{2}
23 1+2.61iT−23T2 1 + 2.61iT - 23T^{2}
29 1+(1.08−0.351i)T+(23.4−17.0i)T2 1 + (1.08 - 0.351i)T + (23.4 - 17.0i)T^{2}
31 1+(3.51−4.84i)T+(−9.57−29.4i)T2 1 + (3.51 - 4.84i)T + (-9.57 - 29.4i)T^{2}
37 1+(−2.69−8.30i)T+(−29.9+21.7i)T2 1 + (-2.69 - 8.30i)T + (-29.9 + 21.7i)T^{2}
41 1+(9.28+3.01i)T+(33.1+24.0i)T2 1 + (9.28 + 3.01i)T + (33.1 + 24.0i)T^{2}
43 1−6.14T+43T2 1 - 6.14T + 43T^{2}
47 1+(−3.31−1.07i)T+(38.0+27.6i)T2 1 + (-3.31 - 1.07i)T + (38.0 + 27.6i)T^{2}
53 1+(−6.63−4.82i)T+(16.3+50.4i)T2 1 + (-6.63 - 4.82i)T + (16.3 + 50.4i)T^{2}
59 1+(−0.712+0.231i)T+(47.7−34.6i)T2 1 + (-0.712 + 0.231i)T + (47.7 - 34.6i)T^{2}
61 1+(3.71+5.11i)T+(−18.8+58.0i)T2 1 + (3.71 + 5.11i)T + (-18.8 + 58.0i)T^{2}
67 1−5.40iT−67T2 1 - 5.40iT - 67T^{2}
71 1+(2.56+3.53i)T+(−21.9+67.5i)T2 1 + (2.56 + 3.53i)T + (-21.9 + 67.5i)T^{2}
73 1+(−0.845+0.274i)T+(59.0−42.9i)T2 1 + (-0.845 + 0.274i)T + (59.0 - 42.9i)T^{2}
79 1+(2.17+1.58i)T+(24.4+75.1i)T2 1 + (2.17 + 1.58i)T + (24.4 + 75.1i)T^{2}
83 1+(12.1−8.85i)T+(25.6−78.9i)T2 1 + (12.1 - 8.85i)T + (25.6 - 78.9i)T^{2}
89 1−4.25T+89T2 1 - 4.25T + 89T^{2}
97 1+(5.92+4.30i)T+(29.9+92.2i)T2 1 + (5.92 + 4.30i)T + (29.9 + 92.2i)T^{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

−15.39530442549097393334868948324, −14.56638391136480686208676572514, −13.54227936732645760471761231135, −12.16800633594278936353688588773, −10.91760408767535257154984502741, −9.924465828474748225584054276566, −8.554776870484596482306572139424, −7.27963159009397202645305742803, −4.13958081293968062323330834959, −3.16150572257945797452774858010, 3.77428715166864530891938463609, 5.73304578858492835897105790577, 7.57628434250880428178986628082, 8.590854788380287971066836602559, 9.162463057963504090476840031582, 11.66884408491895864191720218740, 12.92563757874377212582111775879, 13.99060692433947901166954512860, 15.04525443891162860778575257120, 16.02196323579188038125225044186

Graph of the ZZ-function along the critical line