Properties

Label 2-1344-56.27-c3-0-27
Degree $2$
Conductor $1344$
Sign $0.394 - 0.918i$
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 + 5.86·5-s + (−18.3 − 2.64i)7-s − 9·9-s + 34.5·11-s + 55.8·13-s − 17.5i·15-s + 2.77i·17-s + 67.8i·19-s + (−7.94 + 54.9i)21-s + 176. i·23-s − 90.6·25-s + 27i·27-s − 116. i·29-s − 312.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.524·5-s + (−0.989 − 0.143i)7-s − 0.333·9-s + 0.946·11-s + 1.19·13-s − 0.302i·15-s + 0.0395i·17-s + 0.819i·19-s + (−0.0826 + 0.571i)21-s + 1.59i·23-s − 0.724·25-s + 0.192i·27-s − 0.743i·29-s − 1.80·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.394 - 0.918i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 0.394 - 0.918i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.487913455\)
\(L(\frac12)\) \(\approx\) \(1.487913455\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.3 + 2.64i)T \)
good5 \( 1 - 5.86T + 125T^{2} \)
11 \( 1 - 34.5T + 1.33e3T^{2} \)
13 \( 1 - 55.8T + 2.19e3T^{2} \)
17 \( 1 - 2.77iT - 4.91e3T^{2} \)
19 \( 1 - 67.8iT - 6.85e3T^{2} \)
23 \( 1 - 176. iT - 1.21e4T^{2} \)
29 \( 1 + 116. iT - 2.43e4T^{2} \)
31 \( 1 + 312.T + 2.97e4T^{2} \)
37 \( 1 - 118. iT - 5.06e4T^{2} \)
41 \( 1 + 280. iT - 6.89e4T^{2} \)
43 \( 1 - 15.1T + 7.95e4T^{2} \)
47 \( 1 + 6.34T + 1.03e5T^{2} \)
53 \( 1 - 23.2iT - 1.48e5T^{2} \)
59 \( 1 - 288. iT - 2.05e5T^{2} \)
61 \( 1 + 514.T + 2.26e5T^{2} \)
67 \( 1 + 295.T + 3.00e5T^{2} \)
71 \( 1 - 475. iT - 3.57e5T^{2} \)
73 \( 1 - 473. iT - 3.89e5T^{2} \)
79 \( 1 - 796. iT - 4.93e5T^{2} \)
83 \( 1 - 877. iT - 5.71e5T^{2} \)
89 \( 1 - 33.8iT - 7.04e5T^{2} \)
97 \( 1 + 700. iT - 9.12e5T^{2} \)
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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

−9.376256248671565468156924164088, −8.710148978311649455024727030303, −7.63776142660599079110531612761, −6.89965816365539519855230050913, −5.95208777584569852510455783543, −5.72025931426662082426798194883, −3.93022701714768195988718705945, −3.42473170410419989632640061130, −1.97545674608570855948309360605, −1.12955984260438886323236807223, 0.35605680640433183510891238600, 1.79430077205299728648797933433, 3.07520647400009134175236969006, 3.81707701846976045287095493134, 4.80275928887277779126750889988, 6.02042472658441675336831409292, 6.32412251799233412421084065314, 7.34635449976660200290654857734, 8.739587047186676818633816271816, 9.044728751939392973072805622746

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