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Properties

Label 2-3700-5.4-c1-0-14
Degree $2$
Conductor $3700$
Sign $-0.894 + 0.447i$
Analytic cond. $29.5446$
Root an. cond. $5.43549$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(3700\)    =    \(2^{2} \cdot 5^{2} \cdot 37\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(29.5446\)
Root analytic conductor: \(5.43549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3700} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3700,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.535127736\)
\(L(\frac12)\) \(\approx\) \(1.535127736\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 - iT \)
good3 \( 1 - 3iT - 3T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 - 11iT - 47T^{2} \)
53 \( 1 + 11iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 7iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line