Properties

Label 2-1344-168.53-c0-0-1
Degree $2$
Conductor $1344$
Sign $0.947 + 0.319i$
Analytic cond. $0.670743$
Root an. cond. $0.818989$
Motivic weight $0$
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 + (0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + 1.73i·13-s + (0.866 − 0.5i)19-s − 0.999·21-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (−0.866 + 1.5i)31-s + (1.5 − 0.866i)37-s + (0.866 − 1.49i)39-s i·43-s + (0.499 − 0.866i)49-s − 0.999·57-s + (0.866 + 0.499i)63-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + 1.73i·13-s + (0.866 − 0.5i)19-s − 0.999·21-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (−0.866 + 1.5i)31-s + (1.5 − 0.866i)37-s + (0.866 − 1.49i)39-s i·43-s + (0.499 − 0.866i)49-s − 0.999·57-s + (0.866 + 0.499i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.947 + 0.319i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :0),\ 0.947 + 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9256702973\)
\(L(\frac12)\) \(\approx\) \(0.9256702973\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.830845341724891953825556468489, −8.941838160487469573366830968034, −8.009950433078762756727340570398, −7.08447290103305448515817513961, −6.72893554145078944113405571716, −5.53351083249976991638501681293, −4.76743559030730694256277379358, −4.00273813715112323183398065451, −2.27225170434749405605762787313, −1.21422986512675298244175525548, 1.17469705874623305946668412946, 2.83329241934402238676850798534, 3.92901434078730195800013988621, 5.07275805688660808972814069861, 5.49628187751843572669988334537, 6.30014313573158314903864653018, 7.60501021934564185492426530488, 8.062189299588651202707209724559, 9.271651249613063994119844367179, 9.851780522174119798597175860201

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