Properties

Label 2-1344-56.27-c3-0-20
Degree $2$
Conductor $1344$
Sign $-0.759 - 0.650i$
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 + 1.64·5-s + (15.2 − 10.4i)7-s − 9·9-s − 20.3·11-s + 13.0·13-s + 4.92i·15-s + 23.9i·17-s + 87.7i·19-s + (31.4 + 45.8i)21-s − 73.6i·23-s − 122.·25-s − 27i·27-s − 58.9i·29-s − 124.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.146·5-s + (0.824 − 0.565i)7-s − 0.333·9-s − 0.557·11-s + 0.279·13-s + 0.0847i·15-s + 0.341i·17-s + 1.05i·19-s + (0.326 + 0.476i)21-s − 0.667i·23-s − 0.978·25-s − 0.192i·27-s − 0.377i·29-s − 0.723·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.759 - 0.650i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.759 - 0.650i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.191930337\)
\(L(\frac12)\) \(\approx\) \(1.191930337\)
\(L(\frac{5}{2})\) 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 \( 1 + (-15.2 + 10.4i)T \)
good5 \( 1 - 1.64T + 125T^{2} \)
11 \( 1 + 20.3T + 1.33e3T^{2} \)
13 \( 1 - 13.0T + 2.19e3T^{2} \)
17 \( 1 - 23.9iT - 4.91e3T^{2} \)
19 \( 1 - 87.7iT - 6.85e3T^{2} \)
23 \( 1 + 73.6iT - 1.21e4T^{2} \)
29 \( 1 + 58.9iT - 2.43e4T^{2} \)
31 \( 1 + 124.T + 2.97e4T^{2} \)
37 \( 1 - 56.5iT - 5.06e4T^{2} \)
41 \( 1 - 135. iT - 6.89e4T^{2} \)
43 \( 1 + 259.T + 7.95e4T^{2} \)
47 \( 1 - 217.T + 1.03e5T^{2} \)
53 \( 1 - 529. iT - 1.48e5T^{2} \)
59 \( 1 - 685. iT - 2.05e5T^{2} \)
61 \( 1 - 149.T + 2.26e5T^{2} \)
67 \( 1 - 409.T + 3.00e5T^{2} \)
71 \( 1 - 885. iT - 3.57e5T^{2} \)
73 \( 1 + 269. iT - 3.89e5T^{2} \)
79 \( 1 - 902. iT - 4.93e5T^{2} \)
83 \( 1 - 623. iT - 5.71e5T^{2} \)
89 \( 1 + 986. iT - 7.04e5T^{2} \)
97 \( 1 + 179. iT - 9.12e5T^{2} \)
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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

−9.744299856967247408237560769786, −8.617632847328912895650487208393, −8.062015053869370952418919941603, −7.25419583052147151957309178043, −6.06699763984811911443357196944, −5.36449950767307375097497586171, −4.36725573076353895778963669899, −3.71257176500235302799490641581, −2.39977841161562264415483551900, −1.25302355779875459602839660477, 0.26718651316517011502301288663, 1.65014449366000705964978631954, 2.43148025395089575084666267358, 3.60307989215747981193573186984, 4.96943691020573271923431324580, 5.47274991593613362052243159441, 6.49284124536724728816914726599, 7.40532872622863869352245371085, 8.054699241725022549100793129213, 8.862497796632112375039941587638

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