Properties

Label 2-1344-112.27-c1-0-20
Degree $2$
Conductor $1344$
Sign $0.870 + 0.491i$
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 + (1.76 − 1.76i)5-s + (−0.636 + 2.56i)7-s − 1.00i·9-s + (−0.930 + 0.930i)11-s + (0.170 + 0.170i)13-s − 2.49i·15-s + 4.42i·17-s + (4.04 − 4.04i)19-s + (1.36 + 2.26i)21-s + 9.00·23-s − 1.22i·25-s + (−0.707 − 0.707i)27-s + (7.38 − 7.38i)29-s + 3.11·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.789 − 0.789i)5-s + (−0.240 + 0.970i)7-s − 0.333i·9-s + (−0.280 + 0.280i)11-s + (0.0473 + 0.0473i)13-s − 0.644i·15-s + 1.07i·17-s + (0.928 − 0.928i)19-s + (0.298 + 0.494i)21-s + 1.87·23-s − 0.245i·25-s + (−0.136 − 0.136i)27-s + (1.37 − 1.37i)29-s + 0.559·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.870 + 0.491i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.870 + 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.254535102\)
\(L(\frac12)\) \(\approx\) \(2.254535102\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.636 - 2.56i)T \)
good5 \( 1 + (-1.76 + 1.76i)T - 5iT^{2} \)
11 \( 1 + (0.930 - 0.930i)T - 11iT^{2} \)
13 \( 1 + (-0.170 - 0.170i)T + 13iT^{2} \)
17 \( 1 - 4.42iT - 17T^{2} \)
19 \( 1 + (-4.04 + 4.04i)T - 19iT^{2} \)
23 \( 1 - 9.00T + 23T^{2} \)
29 \( 1 + (-7.38 + 7.38i)T - 29iT^{2} \)
31 \( 1 - 3.11T + 31T^{2} \)
37 \( 1 + (2.72 + 2.72i)T + 37iT^{2} \)
41 \( 1 - 1.02T + 41T^{2} \)
43 \( 1 + (-2.03 + 2.03i)T - 43iT^{2} \)
47 \( 1 + 7.64T + 47T^{2} \)
53 \( 1 + (-3.39 - 3.39i)T + 53iT^{2} \)
59 \( 1 + (2.18 + 2.18i)T + 59iT^{2} \)
61 \( 1 + (4.58 + 4.58i)T + 61iT^{2} \)
67 \( 1 + (-6.50 - 6.50i)T + 67iT^{2} \)
71 \( 1 - 6.26T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 + 6.70iT - 79T^{2} \)
83 \( 1 + (7.18 - 7.18i)T - 83iT^{2} \)
89 \( 1 - 5.14T + 89T^{2} \)
97 \( 1 - 2.08iT - 97T^{2} \)
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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

−9.370090980651297788650840166416, −8.810504965166312242098915026865, −8.148652420458672732924117319276, −7.05129894311482451021735455870, −6.20144953986994692574044130493, −5.38611847046095548675293936882, −4.63494508325299962778180892391, −3.12187472347103648764569116548, −2.28468263813098557180144240691, −1.12294267162694476018401289546, 1.21528800291741534757688519829, 2.89173577871235915244854268383, 3.22877668785646088434973153969, 4.62128200476154254048666976791, 5.40843341015533148773966648050, 6.60258812369253437432615796913, 7.10551633671502810961090345246, 8.038937436513051396115037565778, 9.043667018489798758520177944924, 9.813655152816319197628318598574

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