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Properties

Label 2-1344-7.4-c1-0-26
Degree $2$
Conductor $1344$
Sign $-0.832 + 0.553i$
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 + (−1.60 − 2.77i)5-s + (−1.02 + 2.43i)7-s + (−0.499 − 0.866i)9-s + (2.12 − 3.68i)11-s + 3.15·13-s − 3.20·15-s + (2.20 − 3.81i)17-s + (1.57 + 2.73i)19-s + (1.60 + 2.10i)21-s + (−2.20 − 3.81i)23-s + (−2.62 + 4.54i)25-s − 0.999·27-s − 7.20·29-s + (−1.02 + 1.77i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.715 − 1.23i)5-s + (−0.387 + 0.922i)7-s + (−0.166 − 0.288i)9-s + (0.640 − 1.10i)11-s + 0.874·13-s − 0.826·15-s + (0.533 − 0.924i)17-s + (0.361 + 0.626i)19-s + (0.349 + 0.459i)21-s + (−0.459 − 0.795i)23-s + (−0.524 + 0.909i)25-s − 0.192·27-s − 1.33·29-s + (−0.183 + 0.318i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.832 + 0.553i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.187042434\)
\(L(\frac12)\) \(\approx\) \(1.187042434\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 - 2.43i)T \)
good5 \( 1 + (1.60 + 2.77i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.12 + 3.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.15T + 13T^{2} \)
17 \( 1 + (-2.20 + 3.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.57 - 2.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.20 + 3.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.20T + 29T^{2} \)
31 \( 1 + (1.02 - 1.77i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.82 + 8.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 0.750T + 43T^{2} \)
47 \( 1 + (-1.20 - 2.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.64 - 2.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.12 + 7.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.04 + 7.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.57 + 4.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.40T + 71T^{2} \)
73 \( 1 + (7.62 - 13.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.22 - 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + (1.20 + 2.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−8.966093554980106860033931362889, −8.565404945134111081431049623564, −7.938487484754486883011801143501, −6.83783642898969625244640908604, −5.84177866557102741808066684879, −5.25925001245738222196948468598, −3.89538515294377973594192547882, −3.23499032839407748707885266557, −1.70447062896640533366404696549, −0.48076541119475748237580746525, 1.71285743430930006121464394363, 3.39258147283734611911568689031, 3.60522885662404922096036149103, 4.57125505799470536701346441347, 5.94597434121113239598304846750, 6.92314319886762278871997425661, 7.34463173324986953562543106939, 8.225085989875997618916877547019, 9.283799602861046951262894638988, 10.13415200233085928746640487178

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