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Properties

Label 2-1200-60.47-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.072760838\)
\(L(\frac12)\) \(\approx\) \(1.072760838\)
\(L(1)\) 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

−9.812557536948061461022463574561, −9.452254246984479316148879186376, −8.741782914603429348453313735309, −7.981540232315510327685759725409, −6.80058199737320817471580939049, −5.92011309673505568806087375931, −5.12146938351550802448656829725, −3.90709732369647258909861137397, −3.01609200972239756683414937444, −2.28492571135972876086159921573, 0.874279031667763229495285711211, 2.47595419092273291458371071096, 3.52784041319034434727517903042, 4.12131701679073215332867686006, 5.72985770508222206291467081173, 6.74884801600807357590526138135, 7.09667085380094546767123062037, 7.959703963676752963041471858506, 8.930370203106244238723660490045, 9.707755913366339168086885107773

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