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Properties

Label 2-4600-5.4-c1-0-41
Degree 22
Conductor 46004600
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 36.731136.7311
Root an. cond. 6.060626.06062
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  + 3i·3-s − 2i·7-s − 6·9-s − i·13-s + 6·21-s − i·23-s − 9i·27-s + 3·29-s + 3·31-s − 8i·37-s + 3·39-s + 3·41-s + 2i·43-s − 11i·47-s + 3·49-s + ⋯
L(s)  = 1  + 1.73i·3-s − 0.755i·7-s − 2·9-s − 0.277i·13-s + 1.30·21-s − 0.208i·23-s − 1.73i·27-s + 0.557·29-s + 0.538·31-s − 1.31i·37-s + 0.480·39-s + 0.468·41-s + 0.304i·43-s − 1.60i·47-s + 0.428·49-s + ⋯

Functional equation

Λ(s)=(4600s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4600s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46004600    =    23⋅52⋅232^{3} \cdot 5^{2} \cdot 23
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 36.731136.7311
Root analytic conductor: 6.060626.06062
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4600(4049,⋅)\chi_{4600} (4049, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4600, ( :1/2), 0.447−0.894i)(2,\ 4600,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.7312877541.731287754
L(12)L(\frac12) ≈\approx 1.7312877541.731287754
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
5 1 1
23 1+iT 1 + iT
good3 1−3iT−3T2 1 - 3iT - 3T^{2}
7 1+2iT−7T2 1 + 2iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+iT−13T2 1 + iT - 13T^{2}
17 1−17T2 1 - 17T^{2}
19 1+19T2 1 + 19T^{2}
29 1−3T+29T2 1 - 3T + 29T^{2}
31 1−3T+31T2 1 - 3T + 31T^{2}
37 1+8iT−37T2 1 + 8iT - 37T^{2}
41 1−3T+41T2 1 - 3T + 41T^{2}
43 1−2iT−43T2 1 - 2iT - 43T^{2}
47 1+11iT−47T2 1 + 11iT - 47T^{2}
53 1−14iT−53T2 1 - 14iT - 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1+4T+61T2 1 + 4T + 61T^{2}
67 1+4iT−67T2 1 + 4iT - 67T^{2}
71 1−7T+71T2 1 - 7T + 71T^{2}
73 1−9iT−73T2 1 - 9iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+4iT−83T2 1 + 4iT - 83T^{2}
89 1−2T+89T2 1 - 2T + 89T^{2}
97 1−18iT−97T2 1 - 18iT - 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

−8.662486920376952591533446037602, −7.85205156031396279168738253681, −7.02404820238043737778499722383, −6.04178741119857415136783117697, −5.34321068353149304637587292023, −4.57194838248218024187446697513, −4.01931324585700801304247440971, −3.34227735495195328919822928698, −2.40255080554156027422688666389, −0.69843220659036803517831161922, 0.76528491901308749605670720691, 1.76183632493648659148647100219, 2.49210468537196315234766297334, 3.27879554403493564280515281657, 4.61060177125585013665299574561, 5.51064474057715498178683223258, 6.22224997531827549575503713729, 6.70273101460077679967451138801, 7.45314201601922274236887331474, 8.149097666441831610179014039050

Graph of the ZZ-function along the critical line