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Properties

Label 2-1344-112.27-c1-0-21
Degree $2$
Conductor $1344$
Sign $-0.445 + 0.895i$
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 + (−3.13 + 3.13i)5-s + (−1.14 + 2.38i)7-s − 1.00i·9-s + (−0.422 + 0.422i)11-s + (−3.06 − 3.06i)13-s + 4.43i·15-s − 2.56i·17-s + (−0.955 + 0.955i)19-s + (0.875 + 2.49i)21-s + 5.93·23-s − 14.7i·25-s + (−0.707 − 0.707i)27-s + (1.07 − 1.07i)29-s − 5.77·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−1.40 + 1.40i)5-s + (−0.433 + 0.901i)7-s − 0.333i·9-s + (−0.127 + 0.127i)11-s + (−0.849 − 0.849i)13-s + 1.14i·15-s − 0.623i·17-s + (−0.219 + 0.219i)19-s + (0.191 + 0.544i)21-s + 1.23·23-s − 2.94i·25-s + (−0.136 − 0.136i)27-s + (0.198 − 0.198i)29-s − 1.03·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.445 + 0.895i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.445 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3415518273\)
\(L(\frac12)\) \(\approx\) \(0.3415518273\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 - 2.38i)T \)
good5 \( 1 + (3.13 - 3.13i)T - 5iT^{2} \)
11 \( 1 + (0.422 - 0.422i)T - 11iT^{2} \)
13 \( 1 + (3.06 + 3.06i)T + 13iT^{2} \)
17 \( 1 + 2.56iT - 17T^{2} \)
19 \( 1 + (0.955 - 0.955i)T - 19iT^{2} \)
23 \( 1 - 5.93T + 23T^{2} \)
29 \( 1 + (-1.07 + 1.07i)T - 29iT^{2} \)
31 \( 1 + 5.77T + 31T^{2} \)
37 \( 1 + (3.30 + 3.30i)T + 37iT^{2} \)
41 \( 1 - 6.17T + 41T^{2} \)
43 \( 1 + (-3.35 + 3.35i)T - 43iT^{2} \)
47 \( 1 - 4.38T + 47T^{2} \)
53 \( 1 + (7.85 + 7.85i)T + 53iT^{2} \)
59 \( 1 + (-5.63 - 5.63i)T + 59iT^{2} \)
61 \( 1 + (0.351 + 0.351i)T + 61iT^{2} \)
67 \( 1 + (7.37 + 7.37i)T + 67iT^{2} \)
71 \( 1 - 4.30T + 71T^{2} \)
73 \( 1 + 6.71T + 73T^{2} \)
79 \( 1 + 8.39iT - 79T^{2} \)
83 \( 1 + (6.54 - 6.54i)T - 83iT^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + 3.86iT - 97T^{2} \)
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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

−9.260419637178465689594634612830, −8.409525496261280209719936336696, −7.48348436305122668744362089514, −7.21488210565912685787880478359, −6.26359384202702111778936741889, −5.16351265351716833062604716562, −3.87020754244578849282874023901, −2.97610668191163949838494527319, −2.48864974396243110847080361087, −0.14193057015816538200420121296, 1.27223150376955049936157551367, 3.06973213621163609405545620593, 4.12399459216077895991831446864, 4.44657372868598393591982597542, 5.40326958595693355676967428217, 6.94452183133145652631902622985, 7.51639410957279299152826431759, 8.327779690199411750241295184611, 9.062617856703485798228514319050, 9.589349721331227837735952308072

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