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Properties

Label 2-1344-8.5-c3-0-25
Degree $2$
Conductor $1344$
Sign $-0.258 - 0.965i$
Analytic cond. $79.2985$
Root an. cond. $8.90497$
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  + 3i·3-s + 13.9i·5-s + 7·7-s − 9·9-s − 51.5i·11-s − 30.7i·13-s − 41.9·15-s + 0.246·17-s + 2.72i·19-s + 21i·21-s + 14.5·23-s − 70.6·25-s − 27i·27-s + 167. i·29-s + 198.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.25i·5-s + 0.377·7-s − 0.333·9-s − 1.41i·11-s − 0.656i·13-s − 0.722·15-s + 0.00351·17-s + 0.0328i·19-s + 0.218i·21-s + 0.131·23-s − 0.565·25-s − 0.192i·27-s + 1.06i·29-s + 1.15·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -0.258 - 0.965i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.000203488\)
\(L(\frac12)\) \(\approx\) \(2.000203488\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
7 \( 1 - 7T \)
good5 \( 1 - 13.9iT - 125T^{2} \)
11 \( 1 + 51.5iT - 1.33e3T^{2} \)
13 \( 1 + 30.7iT - 2.19e3T^{2} \)
17 \( 1 - 0.246T + 4.91e3T^{2} \)
19 \( 1 - 2.72iT - 6.85e3T^{2} \)
23 \( 1 - 14.5T + 1.21e4T^{2} \)
29 \( 1 - 167. iT - 2.43e4T^{2} \)
31 \( 1 - 198.T + 2.97e4T^{2} \)
37 \( 1 - 321. iT - 5.06e4T^{2} \)
41 \( 1 - 448.T + 6.89e4T^{2} \)
43 \( 1 - 86.5iT - 7.95e4T^{2} \)
47 \( 1 + 33.5T + 1.03e5T^{2} \)
53 \( 1 - 299. iT - 1.48e5T^{2} \)
59 \( 1 - 197. iT - 2.05e5T^{2} \)
61 \( 1 + 476. iT - 2.26e5T^{2} \)
67 \( 1 - 824. iT - 3.00e5T^{2} \)
71 \( 1 - 503.T + 3.57e5T^{2} \)
73 \( 1 + 1.12e3T + 3.89e5T^{2} \)
79 \( 1 - 684.T + 4.93e5T^{2} \)
83 \( 1 - 1.11e3iT - 5.71e5T^{2} \)
89 \( 1 + 643.T + 7.04e5T^{2} \)
97 \( 1 + 168.T + 9.12e5T^{2} \)
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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

−9.605953636241230387781161732203, −8.572803053912631290867140450027, −7.993508760334724100411484878796, −6.95714933167262221239124334462, −6.13571543491317070666863161372, −5.39743960202451687013631700185, −4.27306615359519797889846290938, −3.16479214053808136201191133457, −2.76612978747388826918665698808, −1.00027398623380914356270062023, 0.54237121857892061867259752738, 1.60075169978143216544837004783, 2.40871610745409324412185912928, 4.14650931148321965258616376408, 4.68175339821648271438113751522, 5.60017447801359916854873862853, 6.60910811405744945002264358336, 7.50747252694148774057549644447, 8.116369950390517349661968143063, 9.067496789320883542636668582715

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