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Properties

Label 2-1344-112.27-c1-0-19
Degree $2$
Conductor $1344$
Sign $0.963 - 0.268i$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)3-s + (1.45 − 1.45i)5-s + (2.11 + 1.59i)7-s − 1.00i·9-s + (2.03 − 2.03i)11-s + (4.25 + 4.25i)13-s + 2.05i·15-s − 2.62i·17-s + (−2.19 + 2.19i)19-s + (−2.61 + 0.369i)21-s − 4.12·23-s + 0.761i·25-s + (0.707 + 0.707i)27-s + (5.04 − 5.04i)29-s + 1.60·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.651 − 0.651i)5-s + (0.798 + 0.601i)7-s − 0.333i·9-s + (0.612 − 0.612i)11-s + (1.18 + 1.18i)13-s + 0.531i·15-s − 0.635i·17-s + (−0.504 + 0.504i)19-s + (−0.571 + 0.0806i)21-s − 0.860·23-s + 0.152i·25-s + (0.136 + 0.136i)27-s + (0.936 − 0.936i)29-s + 0.288·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.963 - 0.268i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.963 - 0.268i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.942259482\)
\(L(\frac12)\) \(\approx\) \(1.942259482\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-2.11 - 1.59i)T \)
good5 \( 1 + (-1.45 + 1.45i)T - 5iT^{2} \)
11 \( 1 + (-2.03 + 2.03i)T - 11iT^{2} \)
13 \( 1 + (-4.25 - 4.25i)T + 13iT^{2} \)
17 \( 1 + 2.62iT - 17T^{2} \)
19 \( 1 + (2.19 - 2.19i)T - 19iT^{2} \)
23 \( 1 + 4.12T + 23T^{2} \)
29 \( 1 + (-5.04 + 5.04i)T - 29iT^{2} \)
31 \( 1 - 1.60T + 31T^{2} \)
37 \( 1 + (6.27 + 6.27i)T + 37iT^{2} \)
41 \( 1 - 2.45T + 41T^{2} \)
43 \( 1 + (-5.72 + 5.72i)T - 43iT^{2} \)
47 \( 1 + 6.44T + 47T^{2} \)
53 \( 1 + (-5.91 - 5.91i)T + 53iT^{2} \)
59 \( 1 + (-5.64 - 5.64i)T + 59iT^{2} \)
61 \( 1 + (-9.13 - 9.13i)T + 61iT^{2} \)
67 \( 1 + (-4.56 - 4.56i)T + 67iT^{2} \)
71 \( 1 - 1.11T + 71T^{2} \)
73 \( 1 + 4.88T + 73T^{2} \)
79 \( 1 + 12.6iT - 79T^{2} \)
83 \( 1 + (7.46 - 7.46i)T - 83iT^{2} \)
89 \( 1 + 8.42T + 89T^{2} \)
97 \( 1 - 7.55iT - 97T^{2} \)
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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

−9.534705759975777159124627132882, −8.741925178267725010789431566337, −8.494054830677387319557292331732, −7.07301100348239318493631896990, −5.96476934364758125956083722694, −5.68887070995038063434464740889, −4.52076692165033767660962776073, −3.85952917095938006568935813799, −2.21488610508147957505195902448, −1.17713295759214940993622647317, 1.10819563439107670414594130140, 2.12092949817144312160316125879, 3.45977942743885828563254757662, 4.52896096386396477248282132488, 5.51840571918101325540998700902, 6.45404056608808197991353200546, 6.86151337882961112949228789408, 8.079430153856647973120781487006, 8.473722802146682158093311923676, 9.874448840177900369723763371298

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