Properties

Label 2-585-13.3-c1-0-19
Degree 22
Conductor 585585
Sign 0.401+0.915i-0.401 + 0.915i
Analytic cond. 4.671244.67124
Root an. cond. 2.161302.16130
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  + (−0.313 − 0.542i)2-s + (0.803 − 1.39i)4-s + 5-s + (−2.21 + 3.83i)7-s − 2.26·8-s + (−0.313 − 0.542i)10-s + (−3.02 − 5.24i)11-s + (2.27 − 2.80i)13-s + 2.77·14-s + (−0.898 − 1.55i)16-s + (2.92 − 5.05i)17-s + (1.80 − 3.12i)19-s + (0.803 − 1.39i)20-s + (−1.89 + 3.28i)22-s + (−1.13 − 1.95i)23-s + ⋯
L(s)  = 1  + (−0.221 − 0.383i)2-s + (0.401 − 0.695i)4-s + 0.447·5-s + (−0.837 + 1.45i)7-s − 0.799·8-s + (−0.0991 − 0.171i)10-s + (−0.913 − 1.58i)11-s + (0.629 − 0.776i)13-s + 0.742·14-s + (−0.224 − 0.389i)16-s + (0.708 − 1.22i)17-s + (0.413 − 0.716i)19-s + (0.179 − 0.311i)20-s + (−0.404 + 0.701i)22-s + (−0.235 − 0.408i)23-s + ⋯

Functional equation

Λ(s)=(585s/2ΓC(s)L(s)=((0.401+0.915i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(585s/2ΓC(s+1/2)L(s)=((0.401+0.915i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 585585    =    325133^{2} \cdot 5 \cdot 13
Sign: 0.401+0.915i-0.401 + 0.915i
Analytic conductor: 4.671244.67124
Root analytic conductor: 2.161302.16130
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ585(406,)\chi_{585} (406, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 585, ( :1/2), 0.401+0.915i)(2,\ 585,\ (\ :1/2),\ -0.401 + 0.915i)

Particular Values

L(1)L(1) \approx 0.6464130.989093i0.646413 - 0.989093i
L(12)L(\frac12) \approx 0.6464130.989093i0.646413 - 0.989093i
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 1T 1 - T
13 1+(2.27+2.80i)T 1 + (-2.27 + 2.80i)T
good2 1+(0.313+0.542i)T+(1+1.73i)T2 1 + (0.313 + 0.542i)T + (-1 + 1.73i)T^{2}
7 1+(2.213.83i)T+(3.56.06i)T2 1 + (2.21 - 3.83i)T + (-3.5 - 6.06i)T^{2}
11 1+(3.02+5.24i)T+(5.5+9.52i)T2 1 + (3.02 + 5.24i)T + (-5.5 + 9.52i)T^{2}
17 1+(2.92+5.05i)T+(8.514.7i)T2 1 + (-2.92 + 5.05i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.80+3.12i)T+(9.516.4i)T2 1 + (-1.80 + 3.12i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.13+1.95i)T+(11.5+19.9i)T2 1 + (1.13 + 1.95i)T + (-11.5 + 19.9i)T^{2}
29 1+(4.04+7.00i)T+(14.5+25.1i)T2 1 + (4.04 + 7.00i)T + (-14.5 + 25.1i)T^{2}
31 16.45T+31T2 1 - 6.45T + 31T^{2}
37 1+(0.8981.55i)T+(18.5+32.0i)T2 1 + (-0.898 - 1.55i)T + (-18.5 + 32.0i)T^{2}
41 1+(3.996.92i)T+(20.5+35.5i)T2 1 + (-3.99 - 6.92i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.245.61i)T+(21.537.2i)T2 1 + (3.24 - 5.61i)T + (-21.5 - 37.2i)T^{2}
47 1+3.22T+47T2 1 + 3.22T + 47T^{2}
53 110.0T+53T2 1 - 10.0T + 53T^{2}
59 1+(1.562.70i)T+(29.551.0i)T2 1 + (1.56 - 2.70i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.22+2.12i)T+(30.552.8i)T2 1 + (-1.22 + 2.12i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.163.75i)T+(33.5+58.0i)T2 1 + (-2.16 - 3.75i)T + (-33.5 + 58.0i)T^{2}
71 1+(4.247.35i)T+(35.561.4i)T2 1 + (4.24 - 7.35i)T + (-35.5 - 61.4i)T^{2}
73 1+0.819T+73T2 1 + 0.819T + 73T^{2}
79 1+8.42T+79T2 1 + 8.42T + 79T^{2}
83 12.35T+83T2 1 - 2.35T + 83T^{2}
89 1+(0.386+0.669i)T+(44.5+77.0i)T2 1 + (0.386 + 0.669i)T + (-44.5 + 77.0i)T^{2}
97 1+(3.16+5.47i)T+(48.584.0i)T2 1 + (-3.16 + 5.47i)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

−10.29320294575057747898372949032, −9.667747676795199349613147421695, −8.888005058639855975631408821325, −7.998141459004318348391460688191, −6.43595641229417738048622737914, −5.80445896143313874752251422607, −5.29528570803006147693409112089, −2.91354768050988957628691420427, −2.73814304254478478460589027174, −0.69843594285636750727066279333, 1.81544647822290738232505139816, 3.38912327447566268442115279191, 4.18099297643239660347377868047, 5.71821933856602873730027077643, 6.78314005838590830084075938010, 7.30081992354382502589821205358, 8.095528496786361692812514600842, 9.294588216893771859651578108735, 10.17846407797431878963646824252, 10.63748168655951209722069251234

Graph of the ZZ-function along the critical line