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Properties

Label 2-1344-48.35-c1-0-38
Degree $2$
Conductor $1344$
Sign $0.134 + 0.990i$
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  + (0.945 + 1.45i)3-s + (−1.69 − 1.69i)5-s − 7-s + (−1.21 + 2.74i)9-s + (0.445 − 0.445i)11-s + (−1.51 − 1.51i)13-s + (0.856 − 4.06i)15-s − 1.39i·17-s + (−0.965 + 0.965i)19-s + (−0.945 − 1.45i)21-s − 6.04i·23-s + 0.748i·25-s + (−5.12 + 0.839i)27-s + (4.93 − 4.93i)29-s − 0.470i·31-s + ⋯
L(s)  = 1  + (0.546 + 0.837i)3-s + (−0.758 − 0.758i)5-s − 0.377·7-s + (−0.403 + 0.914i)9-s + (0.134 − 0.134i)11-s + (−0.419 − 0.419i)13-s + (0.221 − 1.04i)15-s − 0.337i·17-s + (−0.221 + 0.221i)19-s + (−0.206 − 0.316i)21-s − 1.26i·23-s + 0.149i·25-s + (−0.986 + 0.161i)27-s + (0.915 − 0.915i)29-s − 0.0844i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.134 + 0.990i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.134 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.008033798\)
\(L(\frac12)\) \(\approx\) \(1.008033798\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.945 - 1.45i)T \)
7 \( 1 + T \)
good5 \( 1 + (1.69 + 1.69i)T + 5iT^{2} \)
11 \( 1 + (-0.445 + 0.445i)T - 11iT^{2} \)
13 \( 1 + (1.51 + 1.51i)T + 13iT^{2} \)
17 \( 1 + 1.39iT - 17T^{2} \)
19 \( 1 + (0.965 - 0.965i)T - 19iT^{2} \)
23 \( 1 + 6.04iT - 23T^{2} \)
29 \( 1 + (-4.93 + 4.93i)T - 29iT^{2} \)
31 \( 1 + 0.470iT - 31T^{2} \)
37 \( 1 + (-1.47 + 1.47i)T - 37iT^{2} \)
41 \( 1 - 8.35T + 41T^{2} \)
43 \( 1 + (8.97 + 8.97i)T + 43iT^{2} \)
47 \( 1 + 4.84T + 47T^{2} \)
53 \( 1 + (3.42 + 3.42i)T + 53iT^{2} \)
59 \( 1 + (-6.61 + 6.61i)T - 59iT^{2} \)
61 \( 1 + (8.04 + 8.04i)T + 61iT^{2} \)
67 \( 1 + (4.38 - 4.38i)T - 67iT^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 - 14.0iT - 73T^{2} \)
79 \( 1 - 16.5iT - 79T^{2} \)
83 \( 1 + (9.24 + 9.24i)T + 83iT^{2} \)
89 \( 1 + 5.25T + 89T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−9.477212860130141058202679779553, −8.359979664262145445390087772626, −8.303461522751645683616163526315, −7.13148746260481851165329497930, −5.99086930581468254593741185696, −4.88769856219563046032249857895, −4.33442413105742269514298860377, −3.39691121296312255949186788852, −2.37074384278330430292016376783, −0.38854416491551520225231313055, 1.46734475777539083029434181304, 2.81179868778434208109175545424, 3.43972633294426141876831381408, 4.53462387247054356878545733441, 5.95017402070865973909496168335, 6.75297142256589281851427136624, 7.34304857014438931004446969383, 7.993278802215272470222933236465, 8.933101701696920485881080630541, 9.645844334725851085512598330423

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