Properties

Label 2-117-39.17-c2-0-2
Degree 22
Conductor 117117
Sign 0.3240.945i0.324 - 0.945i
Analytic cond. 3.188013.18801
Root an. cond. 1.785501.78550
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.718 + 1.24i)2-s + (0.967 + 1.67i)4-s + 3.30·5-s + (10.5 − 6.06i)7-s − 8.52·8-s + (−2.37 + 4.11i)10-s + (−7.35 + 12.7i)11-s + (3.48 + 12.5i)13-s + 17.4i·14-s + (2.25 − 3.90i)16-s + (19.9 − 11.4i)17-s + (−9.63 + 5.56i)19-s + (3.19 + 5.54i)20-s + (−10.5 − 18.3i)22-s + (3.03 + 1.75i)23-s + ⋯
L(s)  = 1  + (−0.359 + 0.622i)2-s + (0.241 + 0.419i)4-s + 0.661·5-s + (1.50 − 0.866i)7-s − 1.06·8-s + (−0.237 + 0.411i)10-s + (−0.668 + 1.15i)11-s + (0.268 + 0.963i)13-s + 1.24i·14-s + (0.141 − 0.244i)16-s + (1.17 − 0.676i)17-s + (−0.506 + 0.292i)19-s + (0.159 + 0.277i)20-s + (−0.480 − 0.831i)22-s + (0.131 + 0.0761i)23-s + ⋯

Functional equation

Λ(s)=(117s/2ΓC(s)L(s)=((0.3240.945i)Λ(3s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(117s/2ΓC(s+1)L(s)=((0.3240.945i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32133^{2} \cdot 13
Sign: 0.3240.945i0.324 - 0.945i
Analytic conductor: 3.188013.18801
Root analytic conductor: 1.785501.78550
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ117(17,)\chi_{117} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1), 0.3240.945i)(2,\ 117,\ (\ :1),\ 0.324 - 0.945i)

Particular Values

L(32)L(\frac{3}{2}) \approx 1.17713+0.840248i1.17713 + 0.840248i
L(12)L(\frac12) \approx 1.17713+0.840248i1.17713 + 0.840248i
L(2)L(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
13 1+(3.4812.5i)T 1 + (-3.48 - 12.5i)T
good2 1+(0.7181.24i)T+(23.46i)T2 1 + (0.718 - 1.24i)T + (-2 - 3.46i)T^{2}
5 13.30T+25T2 1 - 3.30T + 25T^{2}
7 1+(10.5+6.06i)T+(24.542.4i)T2 1 + (-10.5 + 6.06i)T + (24.5 - 42.4i)T^{2}
11 1+(7.3512.7i)T+(60.5104.i)T2 1 + (7.35 - 12.7i)T + (-60.5 - 104. i)T^{2}
17 1+(19.9+11.4i)T+(144.5250.i)T2 1 + (-19.9 + 11.4i)T + (144.5 - 250. i)T^{2}
19 1+(9.635.56i)T+(180.5312.i)T2 1 + (9.63 - 5.56i)T + (180.5 - 312. i)T^{2}
23 1+(3.031.75i)T+(264.5+458.i)T2 1 + (-3.03 - 1.75i)T + (264.5 + 458. i)T^{2}
29 1+(10.8+6.23i)T+(420.5+728.i)T2 1 + (10.8 + 6.23i)T + (420.5 + 728. i)T^{2}
31 1+29.9iT961T2 1 + 29.9iT - 961T^{2}
37 1+(22.5+13.0i)T+(684.5+1.18e3i)T2 1 + (22.5 + 13.0i)T + (684.5 + 1.18e3i)T^{2}
41 1+(37.9+65.8i)T+(840.51.45e3i)T2 1 + (-37.9 + 65.8i)T + (-840.5 - 1.45e3i)T^{2}
43 1+(4.347.53i)T+(924.5+1.60e3i)T2 1 + (-4.34 - 7.53i)T + (-924.5 + 1.60e3i)T^{2}
47 131.3T+2.20e3T2 1 - 31.3T + 2.20e3T^{2}
53 152.3iT2.80e3T2 1 - 52.3iT - 2.80e3T^{2}
59 1+(20.6+35.6i)T+(1.74e3+3.01e3i)T2 1 + (20.6 + 35.6i)T + (-1.74e3 + 3.01e3i)T^{2}
61 1+(25.143.5i)T+(1.86e3+3.22e3i)T2 1 + (-25.1 - 43.5i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(102.+59.3i)T+(2.24e3+3.88e3i)T2 1 + (102. + 59.3i)T + (2.24e3 + 3.88e3i)T^{2}
71 1+(16.8+29.1i)T+(2.52e3+4.36e3i)T2 1 + (16.8 + 29.1i)T + (-2.52e3 + 4.36e3i)T^{2}
73 15.21iT5.32e3T2 1 - 5.21iT - 5.32e3T^{2}
79 1+39.7T+6.24e3T2 1 + 39.7T + 6.24e3T^{2}
83 1+141.T+6.88e3T2 1 + 141.T + 6.88e3T^{2}
89 1+(11.9+20.7i)T+(3.96e36.85e3i)T2 1 + (-11.9 + 20.7i)T + (-3.96e3 - 6.85e3i)T^{2}
97 1+(43.0+24.8i)T+(4.70e38.14e3i)T2 1 + (-43.0 + 24.8i)T + (4.70e3 - 8.14e3i)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

−13.75145580640151293661189416901, −12.39882946742280462640385673609, −11.42547602190245696011863112783, −10.26429865561453145507132254915, −9.088072462717220562211924655302, −7.73198848019027849280887473022, −7.29031724666143410762043729178, −5.69109280506249585804412492478, −4.25889349466369117048039222329, −2.01410289120608439050958902970, 1.43868524602874116009912958435, 2.85959878445791617715869984083, 5.36687373681717188278096749195, 5.91036910376092960852970102362, 8.035493524072535014172082745962, 8.813709678232729190204876469786, 10.20813169162465585369224981793, 10.90942593561793081400085306334, 11.77454437543789409960041862591, 12.93362456762354824144165408348

Graph of the ZZ-function along the critical line