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Properties

Label 2-1440-5.4-c3-0-72
Degree $2$
Conductor $1440$
Sign $0.178 + 0.983i$
Analytic cond. $84.9627$
Root an. cond. $9.21752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11 − 2i)5-s − 92i·13-s + 104i·17-s + (117 − 44i)25-s + 130·29-s − 396i·37-s − 230·41-s + 343·49-s − 572i·53-s − 830·61-s + (−184 − 1.01e3i)65-s + 592i·73-s + (208 + 1.14e3i)85-s + 1.67e3·89-s − 1.81e3i·97-s + ⋯
L(s)  = 1  + (0.983 − 0.178i)5-s − 1.96i·13-s + 1.48i·17-s + (0.936 − 0.351i)25-s + 0.832·29-s − 1.75i·37-s − 0.876·41-s + 49-s − 1.48i·53-s − 1.74·61-s + (−0.351 − 1.93i)65-s + 0.949i·73-s + (0.265 + 1.45i)85-s + 1.98·89-s − 1.90i·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.178 + 0.983i$
Analytic conductor: \(84.9627\)
Root analytic conductor: \(9.21752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :3/2),\ 0.178 + 0.983i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.335352297\)
\(L(\frac12)\) \(\approx\) \(2.335352297\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11 + 2i)T \)
good7 \( 1 - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 + 92iT - 2.19e3T^{2} \)
17 \( 1 - 104iT - 4.91e3T^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 - 1.21e4T^{2} \)
29 \( 1 - 130T + 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 + 396iT - 5.06e4T^{2} \)
41 \( 1 + 230T + 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 - 1.03e5T^{2} \)
53 \( 1 + 572iT - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 + 830T + 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 592iT - 3.89e5T^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 - 1.67e3T + 7.04e5T^{2} \)
97 \( 1 + 1.81e3iT - 9.12e5T^{2} \)
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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

−8.886388621346417316408307534218, −8.302487456275872774522789583781, −7.43918566237217857947698655768, −6.30165611412243143443785106388, −5.72393850744701288430332696177, −4.99673806143160264075215229592, −3.74614084849373086636683863282, −2.75461304999521101785378732319, −1.69151459256242394577532870725, −0.53515995628464418432660124075, 1.17842050602854067873942581935, 2.18163134816094451808498914948, 3.09343047945337872246443849535, 4.46881877869629376655316573814, 5.07288415754077701735273686599, 6.27980778682208315304315181533, 6.72653752534507400394146930512, 7.59535219715772785118389792195, 8.890496686440826494482025201192, 9.242606567388223089449599647044

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