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Properties

Label 2-3700-5.4-c1-0-14
Degree 22
Conductor 37003700
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 29.544629.5446
Root an. cond. 5.435495.43549
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 + 3i·7-s − 6·9-s + 5·11-s + 2i·13-s − 4i·17-s + 4·19-s − 9·21-s + 6i·23-s − 9i·27-s − 6·29-s − 4·31-s + 15i·33-s + i·37-s − 6·39-s + ⋯
L(s)  = 1  + 1.73i·3-s + 1.13i·7-s − 2·9-s + 1.50·11-s + 0.554i·13-s − 0.970i·17-s + 0.917·19-s − 1.96·21-s + 1.25i·23-s − 1.73i·27-s − 1.11·29-s − 0.718·31-s + 2.61i·33-s + 0.164i·37-s − 0.960·39-s + ⋯

Functional equation

Λ(s)=(3700s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3700s/2 ΓC(s+1/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37003700    =    22⋅52⋅372^{2} \cdot 5^{2} \cdot 37
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 29.544629.5446
Root analytic conductor: 5.435495.43549
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3700(149,⋅)\chi_{3700} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3700, ( :1/2), −0.894+0.447i)(2,\ 3700,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.5351277361.535127736
L(12)L(\frac12) ≈\approx 1.5351277361.535127736
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
37 1−iT 1 - iT
good3 1−3iT−3T2 1 - 3iT - 3T^{2}
7 1−3iT−7T2 1 - 3iT - 7T^{2}
11 1−5T+11T2 1 - 5T + 11T^{2}
13 1−2iT−13T2 1 - 2iT - 13T^{2}
17 1+4iT−17T2 1 + 4iT - 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1−6iT−23T2 1 - 6iT - 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
41 1+9T+41T2 1 + 9T + 41T^{2}
43 1−10iT−43T2 1 - 10iT - 43T^{2}
47 1−11iT−47T2 1 - 11iT - 47T^{2}
53 1+11iT−53T2 1 + 11iT - 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1+8T+61T2 1 + 8T + 61T^{2}
67 1−8iT−67T2 1 - 8iT - 67T^{2}
71 1−3T+71T2 1 - 3T + 71T^{2}
73 1−7iT−73T2 1 - 7iT - 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+9iT−83T2 1 + 9iT - 83T^{2}
89 1−16T+89T2 1 - 16T + 89T^{2}
97 1+12iT−97T2 1 + 12iT - 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

−9.247067626478045514271913854007, −8.630973745296834545337813472953, −7.53723849116246670227748605365, −6.56505660116425478338035465964, −5.65163070993118750506972687412, −5.19809093348276045843727451222, −4.34564168429581661114610950124, −3.57714894755216189061941912308, −2.94803915282220023452925440241, −1.62305666162658064570248079805, 0.46679520155248079724211613104, 1.32709665508464744154261438324, 2.07195983019963108428417745423, 3.41073894207917821460347287942, 4.00648329672614537788040158079, 5.33222587894705257868403150286, 6.13524271919595832263118252842, 6.84887719235403771576611924510, 7.19060392384695012532481959095, 7.940201670874709845002855367032

Graph of the ZZ-function along the critical line