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Properties

Label 2-1344-7.2-c1-0-5
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 + (0.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)11-s − 5·13-s + (1 + 1.73i)17-s + (1.5 − 2.59i)19-s + (−2 + 1.73i)21-s + (−1 + 1.73i)23-s + (2.5 + 4.33i)25-s − 0.999·27-s − 8·29-s + (0.5 + 0.866i)31-s + (−0.999 + 1.73i)33-s + (−2.5 + 4.33i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.188 + 0.981i)7-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s − 1.38·13-s + (0.242 + 0.420i)17-s + (0.344 − 0.596i)19-s + (−0.436 + 0.377i)21-s + (−0.208 + 0.361i)23-s + (0.5 + 0.866i)25-s − 0.192·27-s − 1.48·29-s + (0.0898 + 0.155i)31-s + (−0.174 + 0.301i)33-s + (−0.410 + 0.711i)37-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: $\chi_{1344} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.334773920\)
\(L(\frac12)\) \(\approx\) \(1.334773920\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.623544045426790490577903888814, −9.327192204610342206542252376047, −8.402506962901579822267630841167, −7.52766666798334128286946322584, −6.73972527445456336930401492939, −5.41742556243583669062272362614, −5.04808887037407870963691357974, −3.86550196191454952908828880438, −2.79846294020751685908283088986, −1.83789708011904392128151396776, 0.50672397937442754960786157792, 1.89096655145811944587339621834, 3.08426178623982885467982210603, 4.07735291482026926123599948242, 5.05732713774579975559722602646, 6.10218499239506650608403797781, 7.09112254250509592145889283674, 7.57400900135297583703220430172, 8.355759841630860226547904969084, 9.375359438055230615212704536554

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