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Properties

Label 2-1344-7.4-c1-0-10
Degree $2$
Conductor $1344$
Sign $0.386 - 0.922i$
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 + 0.866i)5-s + (0.5 + 2.59i)7-s + (−0.499 − 0.866i)9-s + (2.5 − 4.33i)11-s − 0.999·15-s + (2 − 3.46i)17-s + (4 + 6.92i)19-s + (−2.5 − 0.866i)21-s + (2 + 3.46i)23-s + (2 − 3.46i)25-s + 0.999·27-s + 5·29-s + (−1.5 + 2.59i)31-s + (2.5 + 4.33i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.166 − 0.288i)9-s + (0.753 − 1.30i)11-s − 0.258·15-s + (0.485 − 0.840i)17-s + (0.917 + 1.58i)19-s + (−0.545 − 0.188i)21-s + (0.417 + 0.722i)23-s + (0.400 − 0.692i)25-s + 0.192·27-s + 0.928·29-s + (−0.269 + 0.466i)31-s + (0.435 + 0.753i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.709937826\)
\(L(\frac12)\) \(\approx\) \(1.709937826\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.5 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.703020070677481457449985748367, −9.042556911183410788568345235482, −8.316785648138920280600102204253, −7.31814192506636318218101810727, −6.10967083808758111012858445136, −5.77572975485209945145753262008, −4.82175612564775978339069600072, −3.51829934973211870491450887760, −2.87041644261094250076542035864, −1.25290607224779742412771481131, 0.887446126835043527403270575874, 1.86381301376214043548103418804, 3.33703807115261084534040867395, 4.57353053222339915448115060498, 5.04818452485235001522858362991, 6.39970955248528460265734472871, 7.00570008998866287603378591154, 7.63989544819864890239188923328, 8.669353540586289696661026013304, 9.498223138964229223370296265003

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