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Properties

Label 2-1344-48.11-c1-0-9
Degree $2$
Conductor $1344$
Sign $-0.533 - 0.845i$
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  + (1.41 + i)3-s + (−0.414 + 0.414i)5-s + 7-s + (1.00 + 2.82i)9-s + (−3.82 − 3.82i)11-s + (−4.41 + 4.41i)13-s + (−1 + 0.171i)15-s + 1.17i·17-s + (−0.414 − 0.414i)19-s + (1.41 + i)21-s + 7.65i·23-s + 4.65i·25-s + (−1.41 + 5.00i)27-s + (3 + 3i)29-s + 6.48i·31-s + ⋯
L(s)  = 1  + (0.816 + 0.577i)3-s + (−0.185 + 0.185i)5-s + 0.377·7-s + (0.333 + 0.942i)9-s + (−1.15 − 1.15i)11-s + (−1.22 + 1.22i)13-s + (−0.258 + 0.0442i)15-s + 0.284i·17-s + (−0.0950 − 0.0950i)19-s + (0.308 + 0.218i)21-s + 1.59i·23-s + 0.931i·25-s + (−0.272 + 0.962i)27-s + (0.557 + 0.557i)29-s + 1.16i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.533 - 0.845i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.533 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.555498831\)
\(L(\frac12)\) \(\approx\) \(1.555498831\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - i)T \)
7 \( 1 - T \)
good5 \( 1 + (0.414 - 0.414i)T - 5iT^{2} \)
11 \( 1 + (3.82 + 3.82i)T + 11iT^{2} \)
13 \( 1 + (4.41 - 4.41i)T - 13iT^{2} \)
17 \( 1 - 1.17iT - 17T^{2} \)
19 \( 1 + (0.414 + 0.414i)T + 19iT^{2} \)
23 \( 1 - 7.65iT - 23T^{2} \)
29 \( 1 + (-3 - 3i)T + 29iT^{2} \)
31 \( 1 - 6.48iT - 31T^{2} \)
37 \( 1 + (-1.82 - 1.82i)T + 37iT^{2} \)
41 \( 1 - 0.343T + 41T^{2} \)
43 \( 1 + (-3.82 + 3.82i)T - 43iT^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + (-5.82 + 5.82i)T - 53iT^{2} \)
59 \( 1 + (10.0 + 10.0i)T + 59iT^{2} \)
61 \( 1 + (-2.41 + 2.41i)T - 61iT^{2} \)
67 \( 1 + (-7 - 7i)T + 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + 5.17iT - 73T^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + (-8.89 + 8.89i)T - 83iT^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.763513818149264544784476946695, −9.102131190354683852639542040322, −8.284135527352705264344096541638, −7.62642165503271749913619972414, −6.84410911417967824106899033991, −5.39434541846945957116655409327, −4.87470439890109722823175340107, −3.69623030310295847715360077361, −2.92040348225390933103447949828, −1.81789575827153725628722680339, 0.54262937597803777668846791187, 2.35934118085590493804656620394, 2.66962057092234190026227808924, 4.28926946209614655461573189420, 4.92864334965224478349472467393, 6.12154035371422460768268169268, 7.19545466412156150810016797859, 7.87759615722013045353248580999, 8.170817438184613239119299032477, 9.349033666311633233360367722118

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