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Properties

Label 2-1344-48.35-c1-0-32
Degree $2$
Conductor $1344$
Sign $0.122 + 0.992i$
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  + (−0.399 − 1.68i)3-s + (2.09 + 2.09i)5-s − 7-s + (−2.68 + 1.34i)9-s + (3.61 − 3.61i)11-s + (−2.99 − 2.99i)13-s + (2.69 − 4.37i)15-s + 2.25i·17-s + (−2.89 + 2.89i)19-s + (0.399 + 1.68i)21-s − 6.47i·23-s + 3.80i·25-s + (3.34 + 3.97i)27-s + (7.29 − 7.29i)29-s − 6.92i·31-s + ⋯
L(s)  = 1  + (−0.230 − 0.973i)3-s + (0.938 + 0.938i)5-s − 0.377·7-s + (−0.893 + 0.449i)9-s + (1.08 − 1.08i)11-s + (−0.830 − 0.830i)13-s + (0.696 − 1.12i)15-s + 0.545i·17-s + (−0.665 + 0.665i)19-s + (0.0872 + 0.367i)21-s − 1.35i·23-s + 0.761i·25-s + (0.643 + 0.765i)27-s + (1.35 − 1.35i)29-s − 1.24i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.122 + 0.992i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.122 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.585336223\)
\(L(\frac12)\) \(\approx\) \(1.585336223\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.399 + 1.68i)T \)
7 \( 1 + T \)
good5 \( 1 + (-2.09 - 2.09i)T + 5iT^{2} \)
11 \( 1 + (-3.61 + 3.61i)T - 11iT^{2} \)
13 \( 1 + (2.99 + 2.99i)T + 13iT^{2} \)
17 \( 1 - 2.25iT - 17T^{2} \)
19 \( 1 + (2.89 - 2.89i)T - 19iT^{2} \)
23 \( 1 + 6.47iT - 23T^{2} \)
29 \( 1 + (-7.29 + 7.29i)T - 29iT^{2} \)
31 \( 1 + 6.92iT - 31T^{2} \)
37 \( 1 + (-2.12 + 2.12i)T - 37iT^{2} \)
41 \( 1 - 7.93T + 41T^{2} \)
43 \( 1 + (-0.598 - 0.598i)T + 43iT^{2} \)
47 \( 1 - 2.35T + 47T^{2} \)
53 \( 1 + (-2.98 - 2.98i)T + 53iT^{2} \)
59 \( 1 + (-1.85 + 1.85i)T - 59iT^{2} \)
61 \( 1 + (-3.40 - 3.40i)T + 61iT^{2} \)
67 \( 1 + (7.71 - 7.71i)T - 67iT^{2} \)
71 \( 1 + 8.99iT - 71T^{2} \)
73 \( 1 + 9.70iT - 73T^{2} \)
79 \( 1 + 5.37iT - 79T^{2} \)
83 \( 1 + (-4.46 - 4.46i)T + 83iT^{2} \)
89 \( 1 + 4.75T + 89T^{2} \)
97 \( 1 - 3.98T + 97T^{2} \)
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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

−9.479843545860897559859819969402, −8.481254598328112450947918109526, −7.76737598213848129130195906208, −6.70776517222810026804161466126, −6.10235482297221941989653502485, −5.84944574049139907566788070865, −4.20451835734905188015954450488, −2.85781130822571708590562063557, −2.25908277619051679221698529041, −0.71180562213289914685224098495, 1.36876039010191568972187580639, 2.66423065589785275299853647166, 4.04089269196398729130496051920, 4.79428120687825221528417930986, 5.34497171201223234978158045723, 6.49771699966893235376601673383, 7.10316000401335965939778851192, 8.664770791731713678305353919443, 9.266517632292100545790565647689, 9.578253694043550089804574116676

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