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Properties

Label 2-1344-56.53-c1-0-18
Degree $2$
Conductor $1344$
Sign $0.197 + 0.980i$
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  + (0.866 + 0.5i)3-s + (−3.79 + 2.18i)5-s + (−1.32 − 2.29i)7-s + (0.499 + 0.866i)9-s + (−1.73 − i)11-s + 5.19i·13-s − 4.37·15-s + (0.791 − 1.37i)17-s + (2.23 − 1.29i)19-s − 2.64i·21-s + (0.456 + 0.791i)23-s + (7.08 − 12.2i)25-s + 0.999i·27-s − 6.92i·29-s + (3.05 − 5.29i)31-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (−1.69 + 0.978i)5-s + (−0.499 − 0.866i)7-s + (0.166 + 0.288i)9-s + (−0.522 − 0.301i)11-s + 1.44i·13-s − 1.13·15-s + (0.191 − 0.332i)17-s + (0.513 − 0.296i)19-s − 0.577i·21-s + (0.0952 + 0.164i)23-s + (1.41 − 2.45i)25-s + 0.192i·27-s − 1.28i·29-s + (0.548 − 0.950i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.197 + 0.980i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6736815802\)
\(L(\frac12)\) \(\approx\) \(0.6736815802\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (1.32 + 2.29i)T \)
good5 \( 1 + (3.79 - 2.18i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.73 + i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + (-0.791 + 1.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.23 + 1.29i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.456 - 0.791i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + (-3.05 + 5.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0825 - 0.0476i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.16T + 41T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 + (3.10 + 5.37i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.37 + 4.83i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.09 + 1.20i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.58 + 4.37i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.43 - 4.29i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.23 + 3.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 + (1.58 + 2.74i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.581951587770389693857950215905, −8.391262205607209040580002243606, −7.82468269486090619901964011690, −7.02563022815693177937029853861, −6.57910268442852609884060992040, −4.89995825004653238733863941616, −3.92414609279844821673687262579, −3.55100142649166352124475626628, −2.46938594990481695517046880546, −0.29329267823173066956648835027, 1.18979159608540690527093942509, 2.99429479974405186582493064039, 3.46819810613864355357615200167, 4.78706382224634475225632619752, 5.38268246260534437660343921981, 6.68070918512881110453534395556, 7.71866690921896589367573789075, 8.138274831254463004690189808280, 8.715930134973128819241081884688, 9.594357817442065179758623993160

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