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Properties

Label 2-1344-12.11-c1-0-47
Degree $2$
Conductor $1344$
Sign $-0.169 - 0.985i$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.292 − 1.70i)3-s − 3.41i·5-s + i·7-s + (−2.82 + i)9-s − 2·11-s − 3.41·13-s + (−5.82 + i)15-s + 4.82i·17-s + 6.24i·19-s + (1.70 − 0.292i)21-s − 4·23-s − 6.65·25-s + (2.53 + 4.53i)27-s − 4.82i·29-s + 1.17i·31-s + ⋯
L(s)  = 1  + (−0.169 − 0.985i)3-s − 1.52i·5-s + 0.377i·7-s + (−0.942 + 0.333i)9-s − 0.603·11-s − 0.946·13-s + (−1.50 + 0.258i)15-s + 1.17i·17-s + 1.43i·19-s + (0.372 − 0.0639i)21-s − 0.834·23-s − 1.33·25-s + (0.487 + 0.872i)27-s − 0.896i·29-s + 0.210i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.169 - 0.985i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.169 - 0.985i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 + 1.70i)T \)
7 \( 1 - iT \)
good5 \( 1 + 3.41iT - 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 - 4.82iT - 17T^{2} \)
19 \( 1 - 6.24iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 4.82iT - 29T^{2} \)
31 \( 1 - 1.17iT - 31T^{2} \)
37 \( 1 - 0.828T + 37T^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 - 10.4iT - 43T^{2} \)
47 \( 1 - 1.65T + 47T^{2} \)
53 \( 1 + 13.3iT - 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 + 9.31iT - 67T^{2} \)
71 \( 1 + 0.343T + 71T^{2} \)
73 \( 1 + 3.17T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 - 7.65iT - 89T^{2} \)
97 \( 1 - 5.31T + 97T^{2} \)
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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

−8.827849607196117527568187836828, −7.986097217233195107298017048000, −7.83706268692227818143846980223, −6.39161237531080202694772070823, −5.66652331217447227057482516942, −5.01547016794962367795629096795, −3.89356503723654693213742965770, −2.32342747717108861923196996452, −1.46764380245854461524408271164, 0, 2.63333968882598232280466810258, 3.01050821019798566437339530084, 4.27443939273224255981444158242, 5.06547156390727929576099099487, 6.05221570108620455451143033584, 7.08148476927719326938603626392, 7.47592097032904248549790762378, 8.767145205340352152101291129611, 9.668867243093190227444713215317

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