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Properties

Label 2-115-5.4-c1-0-9
Degree 22
Conductor 115115
Sign −0.976+0.214i-0.976 + 0.214i
Analytic cond. 0.9182790.918279
Root an. cond. 0.9582690.958269
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  − 2.37i·2-s − 1.95i·3-s − 3.66·4-s + (0.479 + 2.18i)5-s − 4.66·6-s − 2.28i·7-s + 3.95i·8-s − 0.840·9-s + (5.19 − 1.14i)10-s + 1.12·11-s + 7.17i·12-s + 5.95i·13-s − 5.43·14-s + (4.27 − 0.940i)15-s + 2.09·16-s − 5.80i·17-s + ⋯
L(s)  = 1  − 1.68i·2-s − 1.13i·3-s − 1.83·4-s + (0.214 + 0.976i)5-s − 1.90·6-s − 0.863i·7-s + 1.39i·8-s − 0.280·9-s + (1.64 − 0.361i)10-s + 0.338·11-s + 2.07i·12-s + 1.65i·13-s − 1.45·14-s + (1.10 − 0.242i)15-s + 0.523·16-s − 1.40i·17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((−0.976+0.214i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+1/2) L(s)=((−0.976+0.214i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: −0.976+0.214i-0.976 + 0.214i
Analytic conductor: 0.9182790.918279
Root analytic conductor: 0.9582690.958269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ115(24,⋅)\chi_{115} (24, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :1/2), −0.976+0.214i)(2,\ 115,\ (\ :1/2),\ -0.976 + 0.214i)

Particular Values

L(1)L(1) ≈\approx 0.108893−1.00305i0.108893 - 1.00305i
L(12)L(\frac12) ≈\approx 0.108893−1.00305i0.108893 - 1.00305i
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)
bad5 1+(−0.479−2.18i)T 1 + (-0.479 - 2.18i)T
23 1+iT 1 + iT
good2 1+2.37iT−2T2 1 + 2.37iT - 2T^{2}
3 1+1.95iT−3T2 1 + 1.95iT - 3T^{2}
7 1+2.28iT−7T2 1 + 2.28iT - 7T^{2}
11 1−1.12T+11T2 1 - 1.12T + 11T^{2}
13 1−5.95iT−13T2 1 - 5.95iT - 13T^{2}
17 1+5.80iT−17T2 1 + 5.80iT - 17T^{2}
19 1−4.08T+19T2 1 - 4.08T + 19T^{2}
29 1−0.408T+29T2 1 - 0.408T + 29T^{2}
31 1+3.19T+31T2 1 + 3.19T + 31T^{2}
37 1−9.80iT−37T2 1 - 9.80iT - 37T^{2}
41 1−6.27T+41T2 1 - 6.27T + 41T^{2}
43 1−7.75iT−43T2 1 - 7.75iT - 43T^{2}
47 1−6.40iT−47T2 1 - 6.40iT - 47T^{2}
53 1+6.73iT−53T2 1 + 6.73iT - 53T^{2}
59 1−4.75T+59T2 1 - 4.75T + 59T^{2}
61 1+6.33T+61T2 1 + 6.33T + 61T^{2}
67 1+0.283iT−67T2 1 + 0.283iT - 67T^{2}
71 1+13.9T+71T2 1 + 13.9T + 71T^{2}
73 1−9.61iT−73T2 1 - 9.61iT - 73T^{2}
79 1+4.48T+79T2 1 + 4.48T + 79T^{2}
83 1+10.8iT−83T2 1 + 10.8iT - 83T^{2}
89 1+5.68T+89T2 1 + 5.68T + 89T^{2}
97 1+11.0iT−97T2 1 + 11.0iT - 97T^{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

−13.01594372965946494867125097818, −11.68780640141083658914923861794, −11.44687200453004858254690928588, −10.11003503590055307712284487832, −9.299929702622690237451935480432, −7.42847407166445297181581665105, −6.64728380596904064174942250890, −4.32929558696145820560478787902, −2.83555660150489837730819040837, −1.42939676877140458490427123424, 3.99893594056389816868909038101, 5.38005674754705759608137432761, 5.74132543623244504221758195139, 7.63989835367976680665340664785, 8.700506251679590464744058463536, 9.336111704148776990172713578344, 10.51656850836679121150672956589, 12.34488318098791564667308873126, 13.25664624055274704126806561614, 14.58474084123959755208240556738

Graph of the ZZ-function along the critical line