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Properties

Label 2-1344-112.27-c1-0-26
Degree $2$
Conductor $1344$
Sign $0.365 + 0.930i$
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  + (0.707 − 0.707i)3-s + (1.80 − 1.80i)5-s + (1.67 − 2.05i)7-s − 1.00i·9-s + (4.52 − 4.52i)11-s + (2.59 + 2.59i)13-s − 2.54i·15-s + 6.20i·17-s + (−1.43 + 1.43i)19-s + (−0.269 − 2.63i)21-s + 3.79·23-s − 1.48i·25-s + (−0.707 − 0.707i)27-s + (−1.50 + 1.50i)29-s − 4.03·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.805 − 0.805i)5-s + (0.631 − 0.775i)7-s − 0.333i·9-s + (1.36 − 1.36i)11-s + (0.720 + 0.720i)13-s − 0.657i·15-s + 1.50i·17-s + (−0.329 + 0.329i)19-s + (−0.0588 − 0.574i)21-s + 0.791·23-s − 0.297i·25-s + (−0.136 − 0.136i)27-s + (−0.279 + 0.279i)29-s − 0.724·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.365 + 0.930i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.365 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.631219066\)
\(L(\frac12)\) \(\approx\) \(2.631219066\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (-1.67 + 2.05i)T \)
good5 \( 1 + (-1.80 + 1.80i)T - 5iT^{2} \)
11 \( 1 + (-4.52 + 4.52i)T - 11iT^{2} \)
13 \( 1 + (-2.59 - 2.59i)T + 13iT^{2} \)
17 \( 1 - 6.20iT - 17T^{2} \)
19 \( 1 + (1.43 - 1.43i)T - 19iT^{2} \)
23 \( 1 - 3.79T + 23T^{2} \)
29 \( 1 + (1.50 - 1.50i)T - 29iT^{2} \)
31 \( 1 + 4.03T + 31T^{2} \)
37 \( 1 + (1.57 + 1.57i)T + 37iT^{2} \)
41 \( 1 + 9.26T + 41T^{2} \)
43 \( 1 + (4.81 - 4.81i)T - 43iT^{2} \)
47 \( 1 + 4.48T + 47T^{2} \)
53 \( 1 + (-6.91 - 6.91i)T + 53iT^{2} \)
59 \( 1 + (-0.516 - 0.516i)T + 59iT^{2} \)
61 \( 1 + (-6.02 - 6.02i)T + 61iT^{2} \)
67 \( 1 + (6.19 + 6.19i)T + 67iT^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + 3.85T + 73T^{2} \)
79 \( 1 + 5.09iT - 79T^{2} \)
83 \( 1 + (10.0 - 10.0i)T - 83iT^{2} \)
89 \( 1 + 0.264T + 89T^{2} \)
97 \( 1 + 7.14iT - 97T^{2} \)
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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

−9.098552012297960982105491025284, −8.741752372627135343283321701197, −8.109889101822995678354704474350, −6.86044339111232882359537143160, −6.23728174888679836324125173416, −5.37627157591804004021571039728, −4.11629675778934085036292486672, −3.51541464090305474008716800463, −1.62981823375719086285421341055, −1.28744088745944718075094593543, 1.69195931829271644529277215946, 2.55189839790346691884963644671, 3.58276564230772580656337224834, 4.78690193556448778110234630864, 5.47449665661661403215551349095, 6.65122067081899479827608183859, 7.12546443182833305120489124644, 8.352964579277316206907187287401, 9.087102364427668034730725737055, 9.705161367571656259602877688707

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