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Properties

Label 2-1200-60.23-c0-0-3
Degree $2$
Conductor $1200$
Sign $0.755 + 0.655i$
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 + (0.707 + 0.707i)7-s − 1.00i·9-s + (−1.22 − 1.22i)13-s + 1.73·19-s + 1.00·21-s + (−0.707 − 0.707i)27-s + 1.73i·31-s − 1.73·39-s + (−0.707 + 0.707i)43-s + (1.22 − 1.22i)57-s − 61-s + (0.707 − 0.707i)63-s + (−0.707 − 0.707i)67-s − 1.00·81-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (0.707 + 0.707i)7-s − 1.00i·9-s + (−1.22 − 1.22i)13-s + 1.73·19-s + 1.00·21-s + (−0.707 − 0.707i)27-s + 1.73i·31-s − 1.73·39-s + (−0.707 + 0.707i)43-s + (1.22 − 1.22i)57-s − 61-s + (0.707 − 0.707i)63-s + (−0.707 − 0.707i)67-s − 1.00·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :0),\ 0.755 + 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.384373895\)
\(L(\frac12)\) \(\approx\) \(1.384373895\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - 1.73T + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
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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

−9.678645351431711945446934720245, −8.972170220161357476651347445465, −8.046323469800902542401608608645, −7.62342908227919248909818953928, −6.72848108278590541654484716986, −5.51930082534240513141348938022, −4.92431788753481183732645705547, −3.31768618661380884244617420655, −2.63825893309554279608357118557, −1.38555442302089114742771613235, 1.76470960263916003172936311193, 2.91158429638798735460069858219, 4.09364051827906017197568685505, 4.67246992779978742487750157004, 5.58649687624183956304091301249, 7.11721040851947062464364065710, 7.55358849611220376839857970111, 8.419209171535603555998018872969, 9.552914565475188813818325305869, 9.682349519759176192753716408011

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