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Properties

Label 2-1344-28.19-c1-0-29
Degree $2$
Conductor $1344$
Sign $-0.562 + 0.826i$
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  + (−0.5 − 0.866i)3-s + (−0.834 − 0.481i)5-s + (1.20 − 2.35i)7-s + (−0.499 + 0.866i)9-s + (4.74 − 2.74i)11-s − 3.75i·13-s + 0.963i·15-s + (−0.594 + 0.343i)17-s + (−2.44 + 4.22i)19-s + (−2.64 + 0.138i)21-s + (1.07 + 0.620i)23-s + (−2.03 − 3.52i)25-s + 0.999·27-s + 2.48·29-s + (2.41 + 4.18i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.373 − 0.215i)5-s + (0.453 − 0.891i)7-s + (−0.166 + 0.288i)9-s + (1.43 − 0.826i)11-s − 1.04i·13-s + 0.248i·15-s + (−0.144 + 0.0832i)17-s + (−0.560 + 0.969i)19-s + (−0.576 + 0.0302i)21-s + (0.224 + 0.129i)23-s + (−0.407 − 0.705i)25-s + 0.192·27-s + 0.460·29-s + (0.433 + 0.750i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.562 + 0.826i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.562 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.335380266\)
\(L(\frac12)\) \(\approx\) \(1.335380266\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.20 + 2.35i)T \)
good5 \( 1 + (0.834 + 0.481i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.74 + 2.74i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.75iT - 13T^{2} \)
17 \( 1 + (0.594 - 0.343i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.44 - 4.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.07 - 0.620i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.48T + 29T^{2} \)
31 \( 1 + (-2.41 - 4.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.36 - 2.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.42iT - 41T^{2} \)
43 \( 1 - 5.97iT - 43T^{2} \)
47 \( 1 + (-1.80 + 3.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.04 + 3.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.34 + 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.01 + 5.20i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.17 + 4.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (5.76 - 3.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.22 + 0.707i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.543T + 83T^{2} \)
89 \( 1 + (-0.480 - 0.277i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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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

−9.238023360220837174798279958197, −8.208486611537344985720450323711, −7.942650960784437212546532043404, −6.75530394148655356518208747978, −6.21503287646418970357399262049, −5.11956996012935998887662098493, −4.10460365184220301708107000958, −3.32522301713160558345350510906, −1.63551911552543305818212823807, −0.60955976387646397949038975039, 1.59659212990962645742074197661, 2.79501160281235414252442377383, 4.23146688078737309916327696068, 4.51930342930439621574115381977, 5.76681101692868006016218825456, 6.61651108515208505251753577573, 7.29787043713865215211055840933, 8.524525113194288424204627899095, 9.179047198902887631797581202879, 9.647702081339715121623491160813

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