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Properties

Label 2-104-104.83-c1-0-1
Degree 22
Conductor 104104
Sign −0.289−0.957i-0.289 - 0.957i
Analytic cond. 0.8304440.830444
Root an. cond. 0.9112870.911287
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  + (−1 + i)2-s − 3-s − 2i·4-s + (2.54 + 2.54i)5-s + (1 − i)6-s + (−2.54 + 2.54i)7-s + (2 + 2i)8-s − 2·9-s − 5.09·10-s + (1 + i)11-s + 2i·12-s + (2.54 + 2.54i)13-s − 5.09i·14-s + (−2.54 − 2.54i)15-s − 4·16-s − 3i·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 0.577·3-s − i·4-s + (1.14 + 1.14i)5-s + (0.408 − 0.408i)6-s + (−0.963 + 0.963i)7-s + (0.707 + 0.707i)8-s − 0.666·9-s − 1.61·10-s + (0.301 + 0.301i)11-s + 0.577i·12-s + (0.707 + 0.707i)13-s − 1.36i·14-s + (−0.658 − 0.658i)15-s − 16-s − 0.727i·17-s + ⋯

Functional equation

Λ(s)=(104s/2 ΓC(s) L(s)=((−0.289−0.957i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(104s/2 ΓC(s+1/2) L(s)=((−0.289−0.957i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 104104    =    23⋅132^{3} \cdot 13
Sign: −0.289−0.957i-0.289 - 0.957i
Analytic conductor: 0.8304440.830444
Root analytic conductor: 0.9112870.911287
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ104(83,⋅)\chi_{104} (83, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 104, ( :1/2), −0.289−0.957i)(2,\ 104,\ (\ :1/2),\ -0.289 - 0.957i)

Particular Values

L(1)L(1) ≈\approx 0.382697+0.515726i0.382697 + 0.515726i
L(12)L(\frac12) ≈\approx 0.382697+0.515726i0.382697 + 0.515726i
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)
bad2 1+(1−i)T 1 + (1 - i)T
13 1+(−2.54−2.54i)T 1 + (-2.54 - 2.54i)T
good3 1+T+3T2 1 + T + 3T^{2}
5 1+(−2.54−2.54i)T+5iT2 1 + (-2.54 - 2.54i)T + 5iT^{2}
7 1+(2.54−2.54i)T−7iT2 1 + (2.54 - 2.54i)T - 7iT^{2}
11 1+(−1−i)T+11iT2 1 + (-1 - i)T + 11iT^{2}
17 1+3iT−17T2 1 + 3iT - 17T^{2}
19 1+(−2+2i)T−19iT2 1 + (-2 + 2i)T - 19iT^{2}
23 1−5.09T+23T2 1 - 5.09T + 23T^{2}
29 1+5.09iT−29T2 1 + 5.09iT - 29T^{2}
31 1+(5.09+5.09i)T+31iT2 1 + (5.09 + 5.09i)T + 31iT^{2}
37 1+(2.54−2.54i)T−37iT2 1 + (2.54 - 2.54i)T - 37iT^{2}
41 1+(−6+6i)T−41iT2 1 + (-6 + 6i)T - 41iT^{2}
43 1+iT−43T2 1 + iT - 43T^{2}
47 1+(2.54−2.54i)T−47iT2 1 + (2.54 - 2.54i)T - 47iT^{2}
53 1−5.09iT−53T2 1 - 5.09iT - 53T^{2}
59 1+(−8−8i)T+59iT2 1 + (-8 - 8i)T + 59iT^{2}
61 1−61T2 1 - 61T^{2}
67 1+(−3+3i)T−67iT2 1 + (-3 + 3i)T - 67iT^{2}
71 1+(−7.64−7.64i)T+71iT2 1 + (-7.64 - 7.64i)T + 71iT^{2}
73 1+(6+6i)T+73iT2 1 + (6 + 6i)T + 73iT^{2}
79 1+5.09iT−79T2 1 + 5.09iT - 79T^{2}
83 1+(−5+5i)T−83iT2 1 + (-5 + 5i)T - 83iT^{2}
89 1+(2+2i)T+89iT2 1 + (2 + 2i)T + 89iT^{2}
97 1+(7−7i)T−97iT2 1 + (7 - 7i)T - 97iT^{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

−14.25846987770983420492053383154, −13.38253692476843098282926551077, −11.65571896803484622456948625666, −10.78968984458762246601037391337, −9.558937967384012206759723928695, −9.039021026010379074162872165601, −7.03501935581057867994513552218, −6.23515471476455278429499283065, −5.52985957166114308155843123738, −2.56799555875125013026098134348, 1.08173189395811207877629341027, 3.46072327223121112782592454106, 5.36789693522002550494786699393, 6.64000696166601176652651517608, 8.401852584101365089044823613797, 9.296557111346423641802637298505, 10.29775359764292976226402564992, 11.12279472756644437787749760808, 12.65207214052451607068231670570, 13.00369865335117237236138588059

Graph of the ZZ-function along the critical line