Properties

Label 2-675-45.34-c1-0-1
Degree 22
Conductor 675675
Sign 0.6870.726i0.687 - 0.726i
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
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  + (−0.866 − 0.5i)2-s + (−0.500 − 0.866i)4-s + (−2.59 − 1.5i)7-s + 3i·8-s + (−1 + 1.73i)11-s + (−1.73 + i)13-s + (1.5 + 2.59i)14-s + (0.500 − 0.866i)16-s + 4i·17-s + 8·19-s + (1.73 − 0.999i)22-s + (−2.59 + 1.5i)23-s + 1.99·26-s + 3i·28-s + (0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.250 − 0.433i)4-s + (−0.981 − 0.566i)7-s + 1.06i·8-s + (−0.301 + 0.522i)11-s + (−0.480 + 0.277i)13-s + (0.400 + 0.694i)14-s + (0.125 − 0.216i)16-s + 0.970i·17-s + 1.83·19-s + (0.369 − 0.213i)22-s + (−0.541 + 0.312i)23-s + 0.392·26-s + 0.566i·28-s + (0.0928 − 0.160i)29-s + ⋯

Functional equation

Λ(s)=(675s/2ΓC(s)L(s)=((0.6870.726i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2ΓC(s+1/2)L(s)=((0.6870.726i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33523^{3} \cdot 5^{2}
Sign: 0.6870.726i0.687 - 0.726i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(424,)\chi_{675} (424, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), 0.6870.726i)(2,\ 675,\ (\ :1/2),\ 0.687 - 0.726i)

Particular Values

L(1)L(1) \approx 0.529410+0.227808i0.529410 + 0.227808i
L(12)L(\frac12) \approx 0.529410+0.227808i0.529410 + 0.227808i
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)
bad3 1 1
5 1 1
good2 1+(0.866+0.5i)T+(1+1.73i)T2 1 + (0.866 + 0.5i)T + (1 + 1.73i)T^{2}
7 1+(2.59+1.5i)T+(3.5+6.06i)T2 1 + (2.59 + 1.5i)T + (3.5 + 6.06i)T^{2}
11 1+(11.73i)T+(5.59.52i)T2 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2}
13 1+(1.73i)T+(6.511.2i)T2 1 + (1.73 - i)T + (6.5 - 11.2i)T^{2}
17 14iT17T2 1 - 4iT - 17T^{2}
19 18T+19T2 1 - 8T + 19T^{2}
23 1+(2.591.5i)T+(11.519.9i)T2 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2}
29 1+(0.5+0.866i)T+(14.525.1i)T2 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2}
31 1+(15.5+26.8i)T2 1 + (-15.5 + 26.8i)T^{2}
37 14iT37T2 1 - 4iT - 37T^{2}
41 1+(2.54.33i)T+(20.5+35.5i)T2 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2}
43 1+(6.924i)T+(21.5+37.2i)T2 1 + (-6.92 - 4i)T + (21.5 + 37.2i)T^{2}
47 1+(6.06+3.5i)T+(23.5+40.7i)T2 1 + (6.06 + 3.5i)T + (23.5 + 40.7i)T^{2}
53 12iT53T2 1 - 2iT - 53T^{2}
59 1+(712.1i)T+(29.5+51.0i)T2 1 + (-7 - 12.1i)T + (-29.5 + 51.0i)T^{2}
61 1+(3.56.06i)T+(30.552.8i)T2 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.59+1.5i)T+(33.558.0i)T2 1 + (-2.59 + 1.5i)T + (33.5 - 58.0i)T^{2}
71 1+2T+71T2 1 + 2T + 71T^{2}
73 14iT73T2 1 - 4iT - 73T^{2}
79 1+(35.19i)T+(39.568.4i)T2 1 + (3 - 5.19i)T + (-39.5 - 68.4i)T^{2}
83 1+(7.794.5i)T+(41.5+71.8i)T2 1 + (-7.79 - 4.5i)T + (41.5 + 71.8i)T^{2}
89 1+15T+89T2 1 + 15T + 89T^{2}
97 1+(1.73i)T+(48.5+84.0i)T2 1 + (-1.73 - i)T + (48.5 + 84.0i)T^{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

−10.22407999000492943217883549543, −9.887697750403088476177685671307, −9.215358756927379140123313675578, −8.070132340964677675155044062206, −7.24175662027074163879985833183, −6.14821909496671207394796014582, −5.19054047779975664598592125918, −4.03289701584901501325640499204, −2.71976817588939256949472524394, −1.24308645829007819880232275434, 0.42353469952578296085656667924, 2.77109530669204589411747202351, 3.56402857927525764084556600102, 5.03556487538940634643289030534, 6.02317889951852103025632616963, 7.11118042648535941362385945832, 7.73778310422818767144483377660, 8.738211807504768224312620157485, 9.522507313171193712129945389851, 9.920074968257708355999846022191

Graph of the ZZ-function along the critical line