Properties

Label 2-1200-60.47-c0-0-5
Degree $2$
Conductor $1200$
Sign $0.229 + 0.973i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)3-s + (1.41 − 1.41i)7-s + 1.00i·9-s − 2.00·21-s + (0.707 − 0.707i)27-s + (−1.41 − 1.41i)43-s − 3.00i·49-s + 2·61-s + (1.41 + 1.41i)63-s + (−1.41 + 1.41i)67-s − 1.00·81-s + (1.41 + 1.41i)103-s + 2i·109-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (1.41 − 1.41i)7-s + 1.00i·9-s − 2.00·21-s + (0.707 − 0.707i)27-s + (−1.41 − 1.41i)43-s − 3.00i·49-s + 2·61-s + (1.41 + 1.41i)63-s + (−1.41 + 1.41i)67-s − 1.00·81-s + (1.41 + 1.41i)103-s + 2i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9523646389\)
\(L(\frac12)\) \(\approx\) \(0.9523646389\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.21440078686898351557027584875, −8.690326971760936546789527163827, −7.943254919550881968610540621792, −7.29179208913815828792808368247, −6.65009205907508451557026934974, −5.43690937806494213244648617856, −4.76270548427746030080657825137, −3.81653063637242672878343513336, −2.07364776671274661445798946880, −1.03672170738375410860717161310, 1.67850746664169899658497711978, 2.98876520402881796355534311222, 4.35211579153683385784979435733, 5.07473701428459983573087400675, 5.69567846077450316822158419440, 6.57738372855317584120674119835, 7.85466594608480222611769401287, 8.596412715270400328906903894697, 9.288478805347731116282825412905, 10.14613023945727100520028525829

Graph of the $Z$-function along the critical line