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Properties

Label 2-693-231.62-c0-0-3
Degree 22
Conductor 693693
Sign 0.997−0.0746i0.997 - 0.0746i
Analytic cond. 0.3458520.345852
Root an. cond. 0.5880910.588091
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.734 + 0.533i)2-s + (−0.0542 − 0.166i)4-s + (0.951 − 0.309i)7-s + (0.329 − 1.01i)8-s + (−0.987 − 0.156i)11-s + (0.863 + 0.280i)14-s + (0.642 − 0.466i)16-s + (−0.642 − 0.642i)22-s + 1.78i·23-s + (−0.309 + 0.951i)25-s + (−0.103 − 0.142i)28-s + (0.0966 + 0.297i)29-s − 0.346·32-s + (−0.587 − 1.80i)37-s + 1.61i·43-s + (0.0274 + 0.173i)44-s + ⋯
L(s)  = 1  + (0.734 + 0.533i)2-s + (−0.0542 − 0.166i)4-s + (0.951 − 0.309i)7-s + (0.329 − 1.01i)8-s + (−0.987 − 0.156i)11-s + (0.863 + 0.280i)14-s + (0.642 − 0.466i)16-s + (−0.642 − 0.642i)22-s + 1.78i·23-s + (−0.309 + 0.951i)25-s + (−0.103 − 0.142i)28-s + (0.0966 + 0.297i)29-s − 0.346·32-s + (−0.587 − 1.80i)37-s + 1.61i·43-s + (0.0274 + 0.173i)44-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((0.997−0.0746i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s) L(s)=((0.997−0.0746i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 0.997−0.0746i0.997 - 0.0746i
Analytic conductor: 0.3458520.345852
Root analytic conductor: 0.5880910.588091
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ693(62,⋅)\chi_{693} (62, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :0), 0.997−0.0746i)(2,\ 693,\ (\ :0),\ 0.997 - 0.0746i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3699665311.369966531
L(12)L(\frac12) ≈\approx 1.3699665311.369966531
L(1)L(1) 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)
bad3 1 1
7 1+(−0.951+0.309i)T 1 + (-0.951 + 0.309i)T
11 1+(0.987+0.156i)T 1 + (0.987 + 0.156i)T
good2 1+(−0.734−0.533i)T+(0.309+0.951i)T2 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2}
5 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
13 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
17 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
19 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
23 1−1.78iT−T2 1 - 1.78iT - T^{2}
29 1+(−0.0966−0.297i)T+(−0.809+0.587i)T2 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2}
31 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
37 1+(0.587+1.80i)T+(−0.809+0.587i)T2 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2}
41 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
43 1−1.61iT−T2 1 - 1.61iT - T^{2}
47 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
53 1+(0.183−0.253i)T+(−0.309−0.951i)T2 1 + (0.183 - 0.253i)T + (-0.309 - 0.951i)T^{2}
59 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
61 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
67 1+1.61T+T2 1 + 1.61T + T^{2}
71 1+(1.16+1.59i)T+(−0.309+0.951i)T2 1 + (1.16 + 1.59i)T + (-0.309 + 0.951i)T^{2}
73 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
79 1+(0.690−0.951i)T+(−0.309−0.951i)T2 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2}
83 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
89 1+T2 1 + T^{2}
97 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)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

−10.77254866478324440150655696932, −9.880802234945932004551327014305, −8.951383171827479867588831345357, −7.66011984492660116394288667895, −7.33206266450069660727028453804, −5.91119475130221063048095863582, −5.31822418556367670268786928249, −4.47994157932479598312737273952, −3.36715407851992918063285398973, −1.59401119483080453256658171939, 2.07150309078283174174464477932, 2.93546457510629599122432856361, 4.35470421651887605486003172613, 4.89208157100979186150687756097, 5.90320555195122580458230614250, 7.25759176749485244428223225227, 8.297957438868835182634586480974, 8.586522719768385252551770152589, 10.19853779842858046770062617680, 10.72356197103482372967016379335

Graph of the ZZ-function along the critical line