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Properties

Label 2-1344-7.4-c1-0-30
Degree $2$
Conductor $1344$
Sign $-0.605 - 0.795i$
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  + (0.5 − 0.866i)3-s + (−1 − 1.73i)5-s + (−2.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s − 13-s − 1.99·15-s + (0.5 + 0.866i)19-s + (−2 + 1.73i)21-s + (0.500 − 0.866i)25-s − 0.999·27-s − 4·29-s + (−4.5 + 7.79i)31-s + (0.999 + 1.73i)33-s + (1.00 + 5.19i)35-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.447 − 0.774i)5-s + (−0.944 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s − 0.277·13-s − 0.516·15-s + (0.114 + 0.198i)19-s + (−0.436 + 0.377i)21-s + (0.100 − 0.173i)25-s − 0.192·27-s − 0.742·29-s + (−0.808 + 1.39i)31-s + (0.174 + 0.301i)33-s + (0.169 + 0.878i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.605 - 0.795i$
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: \(1\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.605 - 0.795i)\)

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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (4.5 - 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (8 + 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−8.946021065453619476239263809227, −8.366009501478786564022377620215, −7.34306094790659998382162316382, −6.88981117483715539852989262365, −5.75994367144477926262725928795, −4.79925515410611158297109433282, −3.80820691287412543547877362802, −2.84873014442573848270599844740, −1.47128043266629261208765739997, 0, 2.31137945246950741745179691776, 3.26441095245092673564742696620, 3.83092180207419896372703433654, 5.15662579069428321961009531296, 6.00197520102407497620148519531, 6.94428271219848098841953180851, 7.65725667411586337291753654268, 8.622925224689990078637687944563, 9.408658863571527293190856625463

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