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Properties

Label 2-1344-112.27-c1-0-16
Degree $2$
Conductor $1344$
Sign $0.985 - 0.169i$
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  + (−0.707 + 0.707i)3-s + (−1.80 + 1.80i)5-s + (1.67 + 2.05i)7-s − 1.00i·9-s + (4.52 − 4.52i)11-s + (−2.59 − 2.59i)13-s − 2.54i·15-s − 6.20i·17-s + (1.43 − 1.43i)19-s + (−2.63 − 0.269i)21-s + 3.79·23-s − 1.48i·25-s + (0.707 + 0.707i)27-s + (−1.50 + 1.50i)29-s + 4.03·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.805 + 0.805i)5-s + (0.631 + 0.775i)7-s − 0.333i·9-s + (1.36 − 1.36i)11-s + (−0.720 − 0.720i)13-s − 0.657i·15-s − 1.50i·17-s + (0.329 − 0.329i)19-s + (−0.574 − 0.0588i)21-s + 0.791·23-s − 0.297i·25-s + (0.136 + 0.136i)27-s + (−0.279 + 0.279i)29-s + 0.724·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.985 - 0.169i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.985 - 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.364651633\)
\(L(\frac12)\) \(\approx\) \(1.364651633\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-1.67 - 2.05i)T \)
good5 \( 1 + (1.80 - 1.80i)T - 5iT^{2} \)
11 \( 1 + (-4.52 + 4.52i)T - 11iT^{2} \)
13 \( 1 + (2.59 + 2.59i)T + 13iT^{2} \)
17 \( 1 + 6.20iT - 17T^{2} \)
19 \( 1 + (-1.43 + 1.43i)T - 19iT^{2} \)
23 \( 1 - 3.79T + 23T^{2} \)
29 \( 1 + (1.50 - 1.50i)T - 29iT^{2} \)
31 \( 1 - 4.03T + 31T^{2} \)
37 \( 1 + (1.57 + 1.57i)T + 37iT^{2} \)
41 \( 1 - 9.26T + 41T^{2} \)
43 \( 1 + (4.81 - 4.81i)T - 43iT^{2} \)
47 \( 1 - 4.48T + 47T^{2} \)
53 \( 1 + (-6.91 - 6.91i)T + 53iT^{2} \)
59 \( 1 + (0.516 + 0.516i)T + 59iT^{2} \)
61 \( 1 + (6.02 + 6.02i)T + 61iT^{2} \)
67 \( 1 + (6.19 + 6.19i)T + 67iT^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 3.85T + 73T^{2} \)
79 \( 1 + 5.09iT - 79T^{2} \)
83 \( 1 + (-10.0 + 10.0i)T - 83iT^{2} \)
89 \( 1 - 0.264T + 89T^{2} \)
97 \( 1 - 7.14iT - 97T^{2} \)
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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

−9.416039672313946399118083427057, −9.029145659459230388462794841766, −7.945524834630572175705342299748, −7.20976300039515708421796699799, −6.31074171350602393892561157524, −5.40988275667270273719397123432, −4.60026683810843158412296772619, −3.41002218365798135711959124576, −2.76883533242995009254928663054, −0.78876474858740658386334882789, 1.05287844972732442200935287789, 1.94053863295953422723010551802, 4.03123942513195758336221737171, 4.27802166299176955856029567816, 5.22054069394107256113595500119, 6.53795949944520237341438120596, 7.16026648245374982223011774551, 7.87705436491004651772375526674, 8.678413584549233774354618724447, 9.570245156852148087093810435078

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