↑

Properties

Label 2-416-52.31-c1-0-6
Degree 22
Conductor 416416
Sign 0.984−0.176i0.984 - 0.176i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + (−1.58 + 1.58i)5-s + (−1.58 + 1.58i)7-s + 2·9-s + (4.16 − 4.16i)11-s + (3.58 + 0.418i)13-s + (1.58 + 1.58i)15-s + 7.32i·17-s + (1.16 + 1.16i)19-s + (1.58 + 1.58i)21-s + 7.16·23-s − 5i·27-s − 1.16·29-s + (−1.16 − 1.16i)31-s + (−4.16 − 4.16i)33-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.707 + 0.707i)5-s + (−0.597 + 0.597i)7-s + 0.666·9-s + (1.25 − 1.25i)11-s + (0.993 + 0.116i)13-s + (0.408 + 0.408i)15-s + 1.77i·17-s + (0.266 + 0.266i)19-s + (0.345 + 0.345i)21-s + 1.49·23-s − 0.962i·27-s − 0.215·29-s + (−0.208 − 0.208i)31-s + (−0.724 − 0.724i)33-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.984−0.176i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.984−0.176i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.984−0.176i0.984 - 0.176i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(31,⋅)\chi_{416} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.984−0.176i)(2,\ 416,\ (\ :1/2),\ 0.984 - 0.176i)

Particular Values

L(1)L(1) ≈\approx 1.32953+0.118351i1.32953 + 0.118351i
L(12)L(\frac12) ≈\approx 1.32953+0.118351i1.32953 + 0.118351i
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+(−3.58−0.418i)T 1 + (-3.58 - 0.418i)T
good3 1+iT−3T2 1 + iT - 3T^{2}
5 1+(1.58−1.58i)T−5iT2 1 + (1.58 - 1.58i)T - 5iT^{2}
7 1+(1.58−1.58i)T−7iT2 1 + (1.58 - 1.58i)T - 7iT^{2}
11 1+(−4.16+4.16i)T−11iT2 1 + (-4.16 + 4.16i)T - 11iT^{2}
17 1−7.32iT−17T2 1 - 7.32iT - 17T^{2}
19 1+(−1.16−1.16i)T+19iT2 1 + (-1.16 - 1.16i)T + 19iT^{2}
23 1−7.16T+23T2 1 - 7.16T + 23T^{2}
29 1+1.16T+29T2 1 + 1.16T + 29T^{2}
31 1+(1.16+1.16i)T+31iT2 1 + (1.16 + 1.16i)T + 31iT^{2}
37 1+(3.58+3.58i)T+37iT2 1 + (3.58 + 3.58i)T + 37iT^{2}
41 1+(−5.16+5.16i)T−41iT2 1 + (-5.16 + 5.16i)T - 41iT^{2}
43 1+5T+43T2 1 + 5T + 43T^{2}
47 1+(6.74−6.74i)T−47iT2 1 + (6.74 - 6.74i)T - 47iT^{2}
53 1−9.48T+53T2 1 - 9.48T + 53T^{2}
59 1+(4−4i)T−59iT2 1 + (4 - 4i)T - 59iT^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1+(7.32+7.32i)T+67iT2 1 + (7.32 + 7.32i)T + 67iT^{2}
71 1+(1.58+1.58i)T+71iT2 1 + (1.58 + 1.58i)T + 71iT^{2}
73 1+(6+6i)T+73iT2 1 + (6 + 6i)T + 73iT^{2}
79 1+3.48iT−79T2 1 + 3.48iT - 79T^{2}
83 1+(−5.83−5.83i)T+83iT2 1 + (-5.83 - 5.83i)T + 83iT^{2}
89 1+(2.83+2.83i)T+89iT2 1 + (2.83 + 2.83i)T + 89iT^{2}
97 1+(3.83−3.83i)T−97iT2 1 + (3.83 - 3.83i)T - 97iT^{2}
show more
show less
   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

−11.21099385668825825741397662566, −10.62139598513155544565657799330, −9.208069948101888539678208352365, −8.527541852753756860544006291176, −7.42451545023285964706178686726, −6.47447744590984896716212535394, −5.93501140011661582698058221313, −3.93994847228992022849843275917, −3.28760533636346068304922633387, −1.40064698712683332706225680158, 1.11482002652163050123247441336, 3.39044324178106615893187933425, 4.31733002795279148427775541180, 5.00075067304222311909214820900, 6.83132490313818765493130250903, 7.21494608279508074477071064344, 8.688137590907587223340453864870, 9.445178302844407111346128072562, 10.07718615347605479207250082621, 11.28330529111082331356784376077

Graph of the ZZ-function along the critical line