Properties

Label 2-1344-56.3-c1-0-12
Degree $2$
Conductor $1344$
Sign $0.999 + 0.0244i$
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  + (−0.866 − 0.5i)3-s + (1.29 + 2.23i)5-s + (−1.56 − 2.13i)7-s + (0.499 + 0.866i)9-s + (0.195 − 0.339i)11-s + 1.67·13-s − 2.58i·15-s + (0.678 + 0.391i)17-s + (4.19 − 2.42i)19-s + (0.292 + 2.62i)21-s + (−4.47 + 2.58i)23-s + (−0.839 + 1.45i)25-s − 0.999i·27-s + 1.79i·29-s + (1.17 − 2.03i)31-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.577 + 1.00i)5-s + (−0.592 − 0.805i)7-s + (0.166 + 0.288i)9-s + (0.0590 − 0.102i)11-s + 0.465·13-s − 0.667i·15-s + (0.164 + 0.0949i)17-s + (0.962 − 0.555i)19-s + (0.0637 + 0.573i)21-s + (−0.933 + 0.538i)23-s + (−0.167 + 0.290i)25-s − 0.192i·27-s + 0.333i·29-s + (0.211 − 0.365i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.999 + 0.0244i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.999 + 0.0244i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.496042669\)
\(L(\frac12)\) \(\approx\) \(1.496042669\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (1.56 + 2.13i)T \)
good5 \( 1 + (-1.29 - 2.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.195 + 0.339i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.67T + 13T^{2} \)
17 \( 1 + (-0.678 - 0.391i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.19 + 2.42i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.47 - 2.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.79iT - 29T^{2} \)
31 \( 1 + (-1.17 + 2.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.83 + 2.79i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + (-6.59 - 11.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.23 - 4.75i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.132 + 0.0766i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.26 - 3.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.46 + 4.27i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.524iT - 71T^{2} \)
73 \( 1 + (10.2 + 5.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.51 - 1.45i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.3iT - 83T^{2} \)
89 \( 1 + (-10.8 + 6.27i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.0iT - 97T^{2} \)
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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

−9.781613183307796609938869939786, −8.980358055622018589775844034150, −7.49178554002357747935293106958, −7.31215527756723652292463720168, −6.08644093938842427165244805437, −5.92436444723082111129716010099, −4.43771092464900742541156403333, −3.43446172787719204894405300275, −2.42710276736710643460507528456, −0.934925670472034472936855608146, 0.945352440431839622578255212504, 2.30526008167726928227078869854, 3.62382154016853816220881652316, 4.66085719335791406252091143775, 5.65419278653124174176123026110, 5.91370780886601191654849283374, 7.03616982903958934516049510237, 8.254580388222958278850735737685, 8.887351744406822009900877359609, 9.722999897795528074581159818971

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