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Properties

Label 2-5292-9.4-c1-0-23
Degree 22
Conductor 52925292
Sign 0.928+0.371i0.928 + 0.371i
Analytic cond. 42.256842.2568
Root an. cond. 6.500526.50052
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  + (1.94 − 3.37i)5-s + (2.18 + 3.78i)11-s + (−0.792 + 1.37i)13-s + 5.33·17-s − 4.64·19-s + (0.183 − 0.318i)23-s + (−5.07 − 8.79i)25-s + (5.08 + 8.81i)29-s + (1.14 − 1.98i)31-s + 10.8·37-s + (0.690 − 1.19i)41-s + (3.81 + 6.61i)43-s + (−3.80 − 6.58i)47-s + 0.925·53-s + 17.0·55-s + ⋯
L(s)  = 1  + (0.870 − 1.50i)5-s + (0.659 + 1.14i)11-s + (−0.219 + 0.380i)13-s + 1.29·17-s − 1.06·19-s + (0.0383 − 0.0664i)23-s + (−1.01 − 1.75i)25-s + (0.944 + 1.63i)29-s + (0.206 − 0.357i)31-s + 1.78·37-s + (0.107 − 0.186i)41-s + (0.582 + 1.00i)43-s + (−0.554 − 0.961i)47-s + 0.127·53-s + 2.29·55-s + ⋯

Functional equation

Λ(s)=(5292s/2 ΓC(s) L(s)=((0.928+0.371i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5292s/2 ΓC(s+1/2) L(s)=((0.928+0.371i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 52925292    =    22⋅33⋅722^{2} \cdot 3^{3} \cdot 7^{2}
Sign: 0.928+0.371i0.928 + 0.371i
Analytic conductor: 42.256842.2568
Root analytic conductor: 6.500526.50052
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5292(1765,⋅)\chi_{5292} (1765, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5292, ( :1/2), 0.928+0.371i)(2,\ 5292,\ (\ :1/2),\ 0.928 + 0.371i)

Particular Values

L(1)L(1) ≈\approx 2.5899208162.589920816
L(12)L(\frac12) ≈\approx 2.5899208162.589920816
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 1
3 1 1
7 1 1
good5 1+(−1.94+3.37i)T+(−2.5−4.33i)T2 1 + (-1.94 + 3.37i)T + (-2.5 - 4.33i)T^{2}
11 1+(−2.18−3.78i)T+(−5.5+9.52i)T2 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.792−1.37i)T+(−6.5−11.2i)T2 1 + (0.792 - 1.37i)T + (-6.5 - 11.2i)T^{2}
17 1−5.33T+17T2 1 - 5.33T + 17T^{2}
19 1+4.64T+19T2 1 + 4.64T + 19T^{2}
23 1+(−0.183+0.318i)T+(−11.5−19.9i)T2 1 + (-0.183 + 0.318i)T + (-11.5 - 19.9i)T^{2}
29 1+(−5.08−8.81i)T+(−14.5+25.1i)T2 1 + (-5.08 - 8.81i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.14+1.98i)T+(−15.5−26.8i)T2 1 + (-1.14 + 1.98i)T + (-15.5 - 26.8i)T^{2}
37 1−10.8T+37T2 1 - 10.8T + 37T^{2}
41 1+(−0.690+1.19i)T+(−20.5−35.5i)T2 1 + (-0.690 + 1.19i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3.81−6.61i)T+(−21.5+37.2i)T2 1 + (-3.81 - 6.61i)T + (-21.5 + 37.2i)T^{2}
47 1+(3.80+6.58i)T+(−23.5+40.7i)T2 1 + (3.80 + 6.58i)T + (-23.5 + 40.7i)T^{2}
53 1−0.925T+53T2 1 - 0.925T + 53T^{2}
59 1+(0.460−0.797i)T+(−29.5−51.0i)T2 1 + (0.460 - 0.797i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.27+5.67i)T+(−30.5+52.8i)T2 1 + (3.27 + 5.67i)T + (-30.5 + 52.8i)T^{2}
67 1+(7.50−12.9i)T+(−33.5−58.0i)T2 1 + (7.50 - 12.9i)T + (-33.5 - 58.0i)T^{2}
71 1−4.91T+71T2 1 - 4.91T + 71T^{2}
73 1−7.56T+73T2 1 - 7.56T + 73T^{2}
79 1+(0.987+1.71i)T+(−39.5+68.4i)T2 1 + (0.987 + 1.71i)T + (-39.5 + 68.4i)T^{2}
83 1+(−0.253−0.438i)T+(−41.5+71.8i)T2 1 + (-0.253 - 0.438i)T + (-41.5 + 71.8i)T^{2}
89 1−12.2T+89T2 1 - 12.2T + 89T^{2}
97 1+(−4.45−7.71i)T+(−48.5+84.0i)T2 1 + (-4.45 - 7.71i)T + (-48.5 + 84.0i)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

−8.266897219230684081090315462929, −7.51467560706438317602817537490, −6.57086193760047084163211043304, −5.99745402260862591804948063610, −5.06048257605267736581024313889, −4.65906679062242654198244971451, −3.89405363794656840669277229520, −2.55114032450808069368796815980, −1.65876900884437433584702086088, −0.967597710416652193236394051550, 0.864595929329807412601138768252, 2.15560738196045678047673368989, 2.90072491885599893619297567392, 3.50427417011230386875216486448, 4.49962889139576392205181393986, 5.71527150977619267520991763859, 6.14761854583483919240322108383, 6.54148589494891498467609808109, 7.57308337770889711317870029681, 8.063740219149057541313030307357

Graph of the ZZ-function along the critical line