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Properties

Label 2-1344-7.4-c1-0-25
Degree $2$
Conductor $1344$
Sign $-0.701 + 0.712i$
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 + (−1 − 1.73i)5-s + (0.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)11-s + 3·13-s + 1.99·15-s + (−4 + 6.92i)17-s + (−0.5 − 0.866i)19-s + (2 + 1.73i)21-s + (−4 − 6.92i)23-s + (0.500 − 0.866i)25-s + 0.999·27-s − 4·29-s + (−1.5 + 2.59i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.447 − 0.774i)5-s + (0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (0.301 − 0.522i)11-s + 0.832·13-s + 0.516·15-s + (−0.970 + 1.68i)17-s + (−0.114 − 0.198i)19-s + (0.436 + 0.377i)21-s + (−0.834 − 1.44i)23-s + (0.100 − 0.173i)25-s + 0.192·27-s − 0.742·29-s + (−0.269 + 0.466i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7515009427\)
\(L(\frac12)\) \(\approx\) \(0.7515009427\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (4 - 6.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 11T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−9.172113166519054841487368663910, −8.469005548880659302236151477702, −7.994014533155815016459072780417, −6.62028916833309897743837259004, −6.10914929837380516529081993067, −4.81954978578194148802730692113, −4.17040798576159788791799970000, −3.54563688394244776575358547776, −1.67720739595859911487281981000, −0.31936383816843430848460453449, 1.69118349892583740843031587380, 2.73082522977049613844581953526, 3.78811621341316322020898049339, 5.00460755222413266570912439760, 5.85306347965969343388117094085, 6.72353054450456291288257180882, 7.37274939796707780284354138365, 8.184621746800605907009932520743, 9.150596973173432665257755759531, 9.774090370852958948738259721967

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