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Properties

Label 2-416-104.99-c1-0-1
Degree 22
Conductor 416416
Sign −0.267−0.963i-0.267 - 0.963i
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
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  − 2.19·3-s + (0.274 − 0.274i)5-s + (0.352 + 0.352i)7-s + 1.81·9-s + (0.275 − 0.275i)11-s + (−3.16 + 1.72i)13-s + (−0.602 + 0.602i)15-s + 2.40i·17-s + (4.12 + 4.12i)19-s + (−0.773 − 0.773i)21-s − 6.72·23-s + 4.84i·25-s + 2.60·27-s + 8.97i·29-s + (−5.14 + 5.14i)31-s + ⋯
L(s)  = 1  − 1.26·3-s + (0.122 − 0.122i)5-s + (0.133 + 0.133i)7-s + 0.604·9-s + (0.0831 − 0.0831i)11-s + (−0.877 + 0.479i)13-s + (−0.155 + 0.155i)15-s + 0.582i·17-s + (0.946 + 0.946i)19-s + (−0.168 − 0.168i)21-s − 1.40·23-s + 0.969i·25-s + 0.501·27-s + 1.66i·29-s + (−0.923 + 0.923i)31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.267−0.963i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.267−0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.267−0.963i-0.267 - 0.963i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(47,⋅)\chi_{416} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.267−0.963i)(2,\ 416,\ (\ :1/2),\ -0.267 - 0.963i)

Particular Values

L(1)L(1) ≈\approx 0.342688+0.450829i0.342688 + 0.450829i
L(12)L(\frac12) ≈\approx 0.342688+0.450829i0.342688 + 0.450829i
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
13 1+(3.16−1.72i)T 1 + (3.16 - 1.72i)T
good3 1+2.19T+3T2 1 + 2.19T + 3T^{2}
5 1+(−0.274+0.274i)T−5iT2 1 + (-0.274 + 0.274i)T - 5iT^{2}
7 1+(−0.352−0.352i)T+7iT2 1 + (-0.352 - 0.352i)T + 7iT^{2}
11 1+(−0.275+0.275i)T−11iT2 1 + (-0.275 + 0.275i)T - 11iT^{2}
17 1−2.40iT−17T2 1 - 2.40iT - 17T^{2}
19 1+(−4.12−4.12i)T+19iT2 1 + (-4.12 - 4.12i)T + 19iT^{2}
23 1+6.72T+23T2 1 + 6.72T + 23T^{2}
29 1−8.97iT−29T2 1 - 8.97iT - 29T^{2}
31 1+(5.14−5.14i)T−31iT2 1 + (5.14 - 5.14i)T - 31iT^{2}
37 1+(3.81+3.81i)T+37iT2 1 + (3.81 + 3.81i)T + 37iT^{2}
41 1+(−1.19−1.19i)T+41iT2 1 + (-1.19 - 1.19i)T + 41iT^{2}
43 1−5.40iT−43T2 1 - 5.40iT - 43T^{2}
47 1+(−2.83−2.83i)T+47iT2 1 + (-2.83 - 2.83i)T + 47iT^{2}
53 1+10.8iT−53T2 1 + 10.8iT - 53T^{2}
59 1+(0.881−0.881i)T−59iT2 1 + (0.881 - 0.881i)T - 59iT^{2}
61 1−2.65iT−61T2 1 - 2.65iT - 61T^{2}
67 1+(−1.69−1.69i)T+67iT2 1 + (-1.69 - 1.69i)T + 67iT^{2}
71 1+(−9.28+9.28i)T−71iT2 1 + (-9.28 + 9.28i)T - 71iT^{2}
73 1+(1.19−1.19i)T−73iT2 1 + (1.19 - 1.19i)T - 73iT^{2}
79 1−8.08iT−79T2 1 - 8.08iT - 79T^{2}
83 1+(8.49+8.49i)T+83iT2 1 + (8.49 + 8.49i)T + 83iT^{2}
89 1+(−10.2+10.2i)T−89iT2 1 + (-10.2 + 10.2i)T - 89iT^{2}
97 1+(−6.14−6.14i)T+97iT2 1 + (-6.14 - 6.14i)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

−11.55039405883972805371697490925, −10.66751662877699964841325789507, −9.876800951383570600967813766173, −8.837731314396092250665512275322, −7.60941941201450925623255739218, −6.65403950918608777023278420944, −5.61116077586939935857507953762, −5.03057256403443202380235200762, −3.60279034245119838734972068942, −1.65287169304291665074735012794, 0.42563834523289868350937637631, 2.51267259188176278907973266816, 4.26895573726387778020907997908, 5.27656458284055788999626455182, 6.04747327979744636124331427993, 7.09653176924078593375162146477, 7.976720875643199010550589495466, 9.429024051508372228114137436319, 10.15856731658368772762196047204, 11.04073486588699558671104578055

Graph of the ZZ-function along the critical line