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Properties

Label 2-1440-120.77-c1-0-17
Degree $2$
Conductor $1440$
Sign $0.984 + 0.177i$
Analytic cond. $11.4984$
Root an. cond. $3.39093$
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  + (2.03 − 0.929i)5-s + (2.49 + 2.49i)7-s + 3.92·11-s + (−4.55 − 4.55i)13-s + (1.88 − 1.88i)17-s + 4.61·19-s + (0.741 + 0.741i)23-s + (3.27 − 3.78i)25-s + 4.35i·29-s − 9.67·31-s + (7.39 + 2.75i)35-s + (5.39 − 5.39i)37-s + 6.33i·41-s + (0.206 + 0.206i)43-s + (−3.48 + 3.48i)47-s + ⋯
L(s)  = 1  + (0.909 − 0.415i)5-s + (0.942 + 0.942i)7-s + 1.18·11-s + (−1.26 − 1.26i)13-s + (0.457 − 0.457i)17-s + 1.05·19-s + (0.154 + 0.154i)23-s + (0.654 − 0.756i)25-s + 0.809i·29-s − 1.73·31-s + (1.24 + 0.465i)35-s + (0.887 − 0.887i)37-s + 0.989i·41-s + (0.0314 + 0.0314i)43-s + (−0.507 + 0.507i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.984 + 0.177i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.984 + 0.177i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.339762217\)
\(L(\frac12)\) \(\approx\) \(2.339762217\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2.03 + 0.929i)T \)
good7 \( 1 + (-2.49 - 2.49i)T + 7iT^{2} \)
11 \( 1 - 3.92T + 11T^{2} \)
13 \( 1 + (4.55 + 4.55i)T + 13iT^{2} \)
17 \( 1 + (-1.88 + 1.88i)T - 17iT^{2} \)
19 \( 1 - 4.61T + 19T^{2} \)
23 \( 1 + (-0.741 - 0.741i)T + 23iT^{2} \)
29 \( 1 - 4.35iT - 29T^{2} \)
31 \( 1 + 9.67T + 31T^{2} \)
37 \( 1 + (-5.39 + 5.39i)T - 37iT^{2} \)
41 \( 1 - 6.33iT - 41T^{2} \)
43 \( 1 + (-0.206 - 0.206i)T + 43iT^{2} \)
47 \( 1 + (3.48 - 3.48i)T - 47iT^{2} \)
53 \( 1 + (-1.01 + 1.01i)T - 53iT^{2} \)
59 \( 1 + 0.531iT - 59T^{2} \)
61 \( 1 + 3.00iT - 61T^{2} \)
67 \( 1 + (1.28 - 1.28i)T - 67iT^{2} \)
71 \( 1 - 7.61iT - 71T^{2} \)
73 \( 1 + (0.509 - 0.509i)T - 73iT^{2} \)
79 \( 1 - 1.31iT - 79T^{2} \)
83 \( 1 + (-9.85 + 9.85i)T - 83iT^{2} \)
89 \( 1 - 2.91T + 89T^{2} \)
97 \( 1 + (8.11 + 8.11i)T + 97iT^{2} \)
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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

−9.405869006681948609862361297566, −8.921153573058108158320585544634, −7.898727118813929799228711359779, −7.19933001994437216343196042218, −5.95170848623162390708029539850, −5.32976985492237139466744693371, −4.79635934858871543691765751412, −3.26720208801320061726934488991, −2.22502249955873767528831283142, −1.17038177214203482770149623360, 1.31639910483151230967662819048, 2.15258701217118004745134234378, 3.60248464227771773027323023483, 4.49416043488268720506104534698, 5.35137341963319265830980080680, 6.41165284741394220968971566996, 7.14808346898006807946749882362, 7.71110167218130437468219565510, 8.999140646981916059170246119663, 9.561522634538930222627032455153

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