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Properties

Label 2-1344-28.27-c3-0-62
Degree $2$
Conductor $1344$
Sign $0.996 - 0.0794i$
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  + 3·3-s + 4.33i·5-s + (18.4 − 1.47i)7-s + 9·9-s + 21.1i·11-s − 36.7i·13-s + 12.9i·15-s + 46.5i·17-s + 89.7·19-s + (55.3 − 4.41i)21-s − 190. i·23-s + 106.·25-s + 27·27-s − 196.·29-s + 87.5·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.387i·5-s + (0.996 − 0.0794i)7-s + 0.333·9-s + 0.579i·11-s − 0.784i·13-s + 0.223i·15-s + 0.664i·17-s + 1.08·19-s + (0.575 − 0.0458i)21-s − 1.72i·23-s + 0.849·25-s + 0.192·27-s − 1.26·29-s + 0.507·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0794i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.996 - 0.0794i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 0.996 - 0.0794i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.336131345\)
\(L(\frac12)\) \(\approx\) \(3.336131345\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
7 \( 1 + (-18.4 + 1.47i)T \)
good5 \( 1 - 4.33iT - 125T^{2} \)
11 \( 1 - 21.1iT - 1.33e3T^{2} \)
13 \( 1 + 36.7iT - 2.19e3T^{2} \)
17 \( 1 - 46.5iT - 4.91e3T^{2} \)
19 \( 1 - 89.7T + 6.85e3T^{2} \)
23 \( 1 + 190. iT - 1.21e4T^{2} \)
29 \( 1 + 196.T + 2.43e4T^{2} \)
31 \( 1 - 87.5T + 2.97e4T^{2} \)
37 \( 1 - 17.9T + 5.06e4T^{2} \)
41 \( 1 + 374. iT - 6.89e4T^{2} \)
43 \( 1 - 504. iT - 7.95e4T^{2} \)
47 \( 1 - 399.T + 1.03e5T^{2} \)
53 \( 1 - 19.8T + 1.48e5T^{2} \)
59 \( 1 - 74.2T + 2.05e5T^{2} \)
61 \( 1 - 5.16iT - 2.26e5T^{2} \)
67 \( 1 - 276. iT - 3.00e5T^{2} \)
71 \( 1 + 474. iT - 3.57e5T^{2} \)
73 \( 1 - 653. iT - 3.89e5T^{2} \)
79 \( 1 + 586. iT - 4.93e5T^{2} \)
83 \( 1 + 441.T + 5.71e5T^{2} \)
89 \( 1 + 862. iT - 7.04e5T^{2} \)
97 \( 1 - 890. iT - 9.12e5T^{2} \)
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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

−9.130738681631070228094950583364, −8.391890573494941686860852609012, −7.66565922054750216737946569270, −7.04601888301057680926256646984, −5.90826400022166996516473399635, −4.93269540414292056927207606399, −4.10837833791402406049918156314, −2.99393262019413582167926722090, −2.09008735957922657223002023857, −0.912330405323998262672447367871, 0.964519410051020066152440940913, 1.87484029775160388244400827136, 3.08175636246549228793554103984, 4.05610875975870166862430022199, 5.04520914252457713669591875379, 5.69033739018971295558876707354, 7.08195046026744297725575056409, 7.61976417486000605639683872805, 8.505554419861295236887795164219, 9.162180706995836789439130173633

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