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Properties

Label 2-1344-48.35-c1-0-35
Degree $2$
Conductor $1344$
Sign $0.471 + 0.881i$
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  + (1.57 − 0.714i)3-s + (−1.31 − 1.31i)5-s + 7-s + (1.97 − 2.25i)9-s + (−1.23 + 1.23i)11-s + (4.57 + 4.57i)13-s + (−3.02 − 1.13i)15-s − 5.82i·17-s + (2.27 − 2.27i)19-s + (1.57 − 0.714i)21-s + 2.63i·23-s − 1.51i·25-s + (1.50 − 4.97i)27-s + (4.30 − 4.30i)29-s + 5.81i·31-s + ⋯
L(s)  = 1  + (0.910 − 0.412i)3-s + (−0.590 − 0.590i)5-s + 0.377·7-s + (0.659 − 0.751i)9-s + (−0.372 + 0.372i)11-s + (1.26 + 1.26i)13-s + (−0.781 − 0.294i)15-s − 1.41i·17-s + (0.521 − 0.521i)19-s + (0.344 − 0.155i)21-s + 0.549i·23-s − 0.303i·25-s + (0.290 − 0.956i)27-s + (0.799 − 0.799i)29-s + 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.471 + 0.881i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.471 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.262568201\)
\(L(\frac12)\) \(\approx\) \(2.262568201\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 + 0.714i)T \)
7 \( 1 - T \)
good5 \( 1 + (1.31 + 1.31i)T + 5iT^{2} \)
11 \( 1 + (1.23 - 1.23i)T - 11iT^{2} \)
13 \( 1 + (-4.57 - 4.57i)T + 13iT^{2} \)
17 \( 1 + 5.82iT - 17T^{2} \)
19 \( 1 + (-2.27 + 2.27i)T - 19iT^{2} \)
23 \( 1 - 2.63iT - 23T^{2} \)
29 \( 1 + (-4.30 + 4.30i)T - 29iT^{2} \)
31 \( 1 - 5.81iT - 31T^{2} \)
37 \( 1 + (-6.99 + 6.99i)T - 37iT^{2} \)
41 \( 1 + 3.57T + 41T^{2} \)
43 \( 1 + (2.12 + 2.12i)T + 43iT^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + (-0.573 - 0.573i)T + 53iT^{2} \)
59 \( 1 + (-8.42 + 8.42i)T - 59iT^{2} \)
61 \( 1 + (6.80 + 6.80i)T + 61iT^{2} \)
67 \( 1 + (1.90 - 1.90i)T - 67iT^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 + 0.516iT - 73T^{2} \)
79 \( 1 + 3.10iT - 79T^{2} \)
83 \( 1 + (-2.91 - 2.91i)T + 83iT^{2} \)
89 \( 1 + 4.34T + 89T^{2} \)
97 \( 1 - 8.09T + 97T^{2} \)
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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

−9.258128662158001889987651767230, −8.626866823734521249409237833773, −7.961916443616945933966350738630, −7.17269531380470865830201000499, −6.43509544453475216654288578412, −5.01427726763129470218565385822, −4.29252777721519369995436666212, −3.32429929492023765118319699614, −2.14830200605394870810647119947, −0.960095549713995977401335590164, 1.43483681019697165709700340872, 2.97163419744072237241052544740, 3.47180450531509015598254551900, 4.41628233107678471008780350107, 5.57333120355412657142300474002, 6.48768876509629992671113449239, 7.72627326405958924634019318239, 8.146674049091744581412661294148, 8.644966600116997805004309522418, 9.872086435607186041351362266083

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