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Properties

Label 2-1344-48.35-c1-0-36
Degree $2$
Conductor $1344$
Sign $0.220 + 0.975i$
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  + (0.861 − 1.50i)3-s + (1.19 + 1.19i)5-s − 7-s + (−1.51 − 2.58i)9-s + (−1.10 + 1.10i)11-s + (0.418 + 0.418i)13-s + (2.83 − 0.769i)15-s − 4.01i·17-s + (4.91 − 4.91i)19-s + (−0.861 + 1.50i)21-s + 3.26i·23-s − 2.12i·25-s + (−5.19 + 0.0508i)27-s + (4.61 − 4.61i)29-s − 7.31i·31-s + ⋯
L(s)  = 1  + (0.497 − 0.867i)3-s + (0.536 + 0.536i)5-s − 0.377·7-s + (−0.505 − 0.862i)9-s + (−0.332 + 0.332i)11-s + (0.116 + 0.116i)13-s + (0.731 − 0.198i)15-s − 0.974i·17-s + (1.12 − 1.12i)19-s + (−0.187 + 0.327i)21-s + 0.681i·23-s − 0.425i·25-s + (−0.999 + 0.00979i)27-s + (0.857 − 0.857i)29-s − 1.31i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.220 + 0.975i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.969666580\)
\(L(\frac12)\) \(\approx\) \(1.969666580\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.861 + 1.50i)T \)
7 \( 1 + T \)
good5 \( 1 + (-1.19 - 1.19i)T + 5iT^{2} \)
11 \( 1 + (1.10 - 1.10i)T - 11iT^{2} \)
13 \( 1 + (-0.418 - 0.418i)T + 13iT^{2} \)
17 \( 1 + 4.01iT - 17T^{2} \)
19 \( 1 + (-4.91 + 4.91i)T - 19iT^{2} \)
23 \( 1 - 3.26iT - 23T^{2} \)
29 \( 1 + (-4.61 + 4.61i)T - 29iT^{2} \)
31 \( 1 + 7.31iT - 31T^{2} \)
37 \( 1 + (-2.61 + 2.61i)T - 37iT^{2} \)
41 \( 1 - 8.46T + 41T^{2} \)
43 \( 1 + (3.91 + 3.91i)T + 43iT^{2} \)
47 \( 1 - 8.08T + 47T^{2} \)
53 \( 1 + (-1.78 - 1.78i)T + 53iT^{2} \)
59 \( 1 + (5.55 - 5.55i)T - 59iT^{2} \)
61 \( 1 + (-0.325 - 0.325i)T + 61iT^{2} \)
67 \( 1 + (-1.41 + 1.41i)T - 67iT^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 - 2.89iT - 73T^{2} \)
79 \( 1 - 15.6iT - 79T^{2} \)
83 \( 1 + (9.34 + 9.34i)T + 83iT^{2} \)
89 \( 1 + 6.70T + 89T^{2} \)
97 \( 1 + 3.43T + 97T^{2} \)
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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

−9.524214835172354154084784864162, −8.660049774430664406981258223503, −7.52209674943379347072721031638, −7.18680631183134834791386093063, −6.23793028192970768888198519904, −5.50132450708711904322412803201, −4.16630280632403876030653409030, −2.81737268682986666686559190951, −2.42643034290320295858838760052, −0.809284716332509335805928050150, 1.45011178888582168149035983356, 2.87111930109845668511974560542, 3.64031193733907615819340192210, 4.74140014186049701017763173570, 5.50862195534927114864024819093, 6.26921270271551262086609280527, 7.58639896847769217300021827601, 8.392987779059130034686514934214, 8.998584793276416941462205810719, 9.787193481480361441772288041304

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