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Properties

Label 2-1200-15.14-c2-0-1
Degree $2$
Conductor $1200$
Sign $-0.984 + 0.178i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.79 − 2.40i)3-s + 10.2i·7-s + (−2.53 + 8.63i)9-s + 8.19i·11-s + 13.5i·13-s + 15.4·17-s − 25.4·19-s + (24.5 − 18.3i)21-s − 17.9·23-s + (25.2 − 9.44i)27-s − 42.0i·29-s − 38.4·31-s + (19.6 − 14.7i)33-s + 11.8i·37-s + (32.6 − 24.4i)39-s + ⋯
L(s)  = 1  + (−0.599 − 0.800i)3-s + 1.45i·7-s + (−0.281 + 0.959i)9-s + 0.744i·11-s + 1.04i·13-s + 0.910·17-s − 1.34·19-s + (1.16 − 0.874i)21-s − 0.778·23-s + (0.936 − 0.349i)27-s − 1.44i·29-s − 1.24·31-s + (0.596 − 0.446i)33-s + 0.319i·37-s + (0.836 − 0.626i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.178i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.984 + 0.178i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.984 + 0.178i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.09940631990\)
\(L(\frac12)\) \(\approx\) \(0.09940631990\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.79 + 2.40i)T \)
5 \( 1 \)
good7 \( 1 - 10.2iT - 49T^{2} \)
11 \( 1 - 8.19iT - 121T^{2} \)
13 \( 1 - 13.5iT - 169T^{2} \)
17 \( 1 - 15.4T + 289T^{2} \)
19 \( 1 + 25.4T + 361T^{2} \)
23 \( 1 + 17.9T + 529T^{2} \)
29 \( 1 + 42.0iT - 841T^{2} \)
31 \( 1 + 38.4T + 961T^{2} \)
37 \( 1 - 11.8iT - 1.36e3T^{2} \)
41 \( 1 + 46.3iT - 1.68e3T^{2} \)
43 \( 1 + 54.0iT - 1.84e3T^{2} \)
47 \( 1 + 43.0T + 2.20e3T^{2} \)
53 \( 1 - 82.7T + 2.80e3T^{2} \)
59 \( 1 + 45.8iT - 3.48e3T^{2} \)
61 \( 1 + 93.6T + 3.72e3T^{2} \)
67 \( 1 + 34.4iT - 4.48e3T^{2} \)
71 \( 1 - 68.0iT - 5.04e3T^{2} \)
73 \( 1 - 44.7iT - 5.32e3T^{2} \)
79 \( 1 + 11.7T + 6.24e3T^{2} \)
83 \( 1 - 144.T + 6.88e3T^{2} \)
89 \( 1 + 63.7iT - 7.92e3T^{2} \)
97 \( 1 - 63.9iT - 9.40e3T^{2} \)
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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

−9.976595986331034397028287308924, −9.071417158875165997673794947643, −8.336050430406643575080398162083, −7.46535713824951870806918972132, −6.55669751323525222986759871090, −5.87767725781861319790399338960, −5.12935293964878823007231065185, −4.00591042422996644247355319478, −2.31519104765211491093969922770, −1.87163614779058000021447205667, 0.03414108059897466442791895846, 1.15175485625394893951277889061, 3.18681007261618406866566468637, 3.84337994626564448251712630932, 4.76973317270902699562942846680, 5.71018602352636837578010836444, 6.47215513077254645248665434246, 7.50507419140784593091802854953, 8.290409657732224422741304845646, 9.290273649536580403493325425418

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