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Properties

Label 2-1344-48.11-c1-0-19
Degree $2$
Conductor $1344$
Sign $0.544 + 0.838i$
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  + (−1.71 + 0.223i)3-s + (−2.27 + 2.27i)5-s − 7-s + (2.90 − 0.766i)9-s + (−1.53 − 1.53i)11-s + (−3.63 + 3.63i)13-s + (3.39 − 4.40i)15-s + 1.81i·17-s + (−4.98 − 4.98i)19-s + (1.71 − 0.223i)21-s + 7.21i·23-s − 5.30i·25-s + (−4.81 + 1.96i)27-s + (2.77 + 2.77i)29-s − 4.14i·31-s + ⋯
L(s)  = 1  + (−0.991 + 0.128i)3-s + (−1.01 + 1.01i)5-s − 0.377·7-s + (0.966 − 0.255i)9-s + (−0.463 − 0.463i)11-s + (−1.00 + 1.00i)13-s + (0.876 − 1.13i)15-s + 0.441i·17-s + (−1.14 − 1.14i)19-s + (0.374 − 0.0486i)21-s + 1.50i·23-s − 1.06i·25-s + (−0.925 + 0.377i)27-s + (0.514 + 0.514i)29-s − 0.744i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.544 + 0.838i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.544 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3163273084\)
\(L(\frac12)\) \(\approx\) \(0.3163273084\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 - 0.223i)T \)
7 \( 1 + T \)
good5 \( 1 + (2.27 - 2.27i)T - 5iT^{2} \)
11 \( 1 + (1.53 + 1.53i)T + 11iT^{2} \)
13 \( 1 + (3.63 - 3.63i)T - 13iT^{2} \)
17 \( 1 - 1.81iT - 17T^{2} \)
19 \( 1 + (4.98 + 4.98i)T + 19iT^{2} \)
23 \( 1 - 7.21iT - 23T^{2} \)
29 \( 1 + (-2.77 - 2.77i)T + 29iT^{2} \)
31 \( 1 + 4.14iT - 31T^{2} \)
37 \( 1 + (1.77 + 1.77i)T + 37iT^{2} \)
41 \( 1 + 0.517T + 41T^{2} \)
43 \( 1 + (-4.67 + 4.67i)T - 43iT^{2} \)
47 \( 1 - 1.19T + 47T^{2} \)
53 \( 1 + (-7.94 + 7.94i)T - 53iT^{2} \)
59 \( 1 + (-1.94 - 1.94i)T + 59iT^{2} \)
61 \( 1 + (-2.94 + 2.94i)T - 61iT^{2} \)
67 \( 1 + (-7.85 - 7.85i)T + 67iT^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + 15.7iT - 73T^{2} \)
79 \( 1 - 5.40iT - 79T^{2} \)
83 \( 1 + (7.12 - 7.12i)T - 83iT^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−9.684986745872672466757504921068, −8.704687660981269121768194635802, −7.48764237755091209774763178823, −7.04311247117149073551290644678, −6.33538922087797003397868790907, −5.29281109304886370593474411522, −4.30242648180851143778458292019, −3.53229415414173255289603318460, −2.27160150707184976935173562996, −0.21369977384183452953576171391, 0.805078857584069612535669942390, 2.50400844571143524906925339331, 4.05723487122536300350034617073, 4.69273604276582536164612524255, 5.42361496311725785656938722966, 6.42713142010342825222624401051, 7.36815079717849218753065655956, 8.033987207984683698193426087734, 8.781841691839522667083729223959, 10.17830390189433192045975721100

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