Properties

Label 2-1344-48.11-c1-0-5
Degree $2$
Conductor $1344$
Sign $-0.461 - 0.886i$
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  + (−1.29 + 1.15i)3-s + (0.186 − 0.186i)5-s − 7-s + (0.350 − 2.97i)9-s + (−4.29 − 4.29i)11-s + (2.73 − 2.73i)13-s + (−0.0267 + 0.455i)15-s + 4.98i·17-s + (3.09 + 3.09i)19-s + (1.29 − 1.15i)21-s + 3.55i·23-s + 4.93i·25-s + (2.97 + 4.26i)27-s + (3.75 + 3.75i)29-s + 6.58i·31-s + ⋯
L(s)  = 1  + (−0.747 + 0.664i)3-s + (0.0833 − 0.0833i)5-s − 0.377·7-s + (0.116 − 0.993i)9-s + (−1.29 − 1.29i)11-s + (0.759 − 0.759i)13-s + (−0.00691 + 0.117i)15-s + 1.20i·17-s + (0.710 + 0.710i)19-s + (0.282 − 0.251i)21-s + 0.742i·23-s + 0.986i·25-s + (0.572 + 0.819i)27-s + (0.697 + 0.697i)29-s + 1.18i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.461 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.461 - 0.886i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.461 - 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7186411813\)
\(L(\frac12)\) \(\approx\) \(0.7186411813\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 - 1.15i)T \)
7 \( 1 + T \)
good5 \( 1 + (-0.186 + 0.186i)T - 5iT^{2} \)
11 \( 1 + (4.29 + 4.29i)T + 11iT^{2} \)
13 \( 1 + (-2.73 + 2.73i)T - 13iT^{2} \)
17 \( 1 - 4.98iT - 17T^{2} \)
19 \( 1 + (-3.09 - 3.09i)T + 19iT^{2} \)
23 \( 1 - 3.55iT - 23T^{2} \)
29 \( 1 + (-3.75 - 3.75i)T + 29iT^{2} \)
31 \( 1 - 6.58iT - 31T^{2} \)
37 \( 1 + (4.82 + 4.82i)T + 37iT^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 + (-5.79 + 5.79i)T - 43iT^{2} \)
47 \( 1 + 3.22T + 47T^{2} \)
53 \( 1 + (7.95 - 7.95i)T - 53iT^{2} \)
59 \( 1 + (-3.18 - 3.18i)T + 59iT^{2} \)
61 \( 1 + (3.17 - 3.17i)T - 61iT^{2} \)
67 \( 1 + (2.39 + 2.39i)T + 67iT^{2} \)
71 \( 1 - 9.06iT - 71T^{2} \)
73 \( 1 + 2.32iT - 73T^{2} \)
79 \( 1 - 6.89iT - 79T^{2} \)
83 \( 1 + (-1.12 + 1.12i)T - 83iT^{2} \)
89 \( 1 - 7.81T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−10.26044357601939046551776193548, −9.019722232164932883575252050321, −8.430245729506213772176343578059, −7.45509682243198153119311411176, −6.27706270016015643210262839224, −5.59743057310609090799194245927, −5.15606800019648184265554713654, −3.57334693380451746760576090168, −3.26708682344010419360605261699, −1.21438068366085880523198592734, 0.35653746195376890190581684677, 1.97759587486153750887752544796, 2.87506086839896757684988648466, 4.59167687509130083367957207888, 5.02203876116084642592064161912, 6.24107617691760578267336311867, 6.80805722488859487028566081838, 7.58188361188166545804477803134, 8.347118108308398035782977372281, 9.582134308735634342820857189673

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