Properties

Label 2-1344-168.83-c0-0-1
Degree $2$
Conductor $1344$
Sign $-0.707 - 0.707i$
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  − 3-s + 2i·5-s + i·7-s + 9-s − 2i·15-s i·21-s − 3·25-s − 27-s − 2·35-s + 2i·45-s − 49-s + 2·59-s + i·63-s + 3·75-s + 2i·79-s + ⋯
L(s)  = 1  − 3-s + 2i·5-s + i·7-s + 9-s − 2i·15-s i·21-s − 3·25-s − 27-s − 2·35-s + 2i·45-s − 49-s + 2·59-s + i·63-s + 3·75-s + 2i·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6825217664\)
\(L(\frac12)\) \(\approx\) \(0.6825217664\)
\(L(1)\) 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 \( 1 + T \)
7 \( 1 - iT \)
good5 \( 1 - 2iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 2T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.12009930328191419208504492812, −9.716622289198695990674773579882, −8.408509846240292794695720495208, −7.34337088562514596514264248598, −6.77216262394216124770656821525, −6.03489072500338395833203482049, −5.40788031447470292434097615155, −4.07232481492294532366636603482, −3.01104115231651150622631176805, −2.05481371925434269775501931639, 0.67180858255211396267284239383, 1.65194193267853584179357215367, 3.86257621589424879599621546664, 4.52948100835376894732925002812, 5.20711583231113957452755109248, 5.99152847418437600599608673085, 7.06875265215283227726168899678, 7.895890369083974894057273419673, 8.715763664452422322788896601995, 9.617482801060288170028143714073

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