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Properties

Label 2-1344-7.4-c1-0-28
Degree $2$
Conductor $1344$
Sign $-0.928 + 0.371i$
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 + (−2.13 − 3.70i)5-s + (1.5 − 2.17i)7-s + (−0.499 − 0.866i)9-s + (2.13 − 3.70i)11-s + 1.27·13-s − 4.27·15-s + (2 − 3.46i)17-s + (−0.637 − 1.10i)19-s + (−1.13 − 2.38i)21-s + (2 + 3.46i)23-s + (−6.63 + 11.4i)25-s − 0.999·27-s + 2.27·29-s + (−0.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.955 − 1.65i)5-s + (0.566 − 0.823i)7-s + (−0.166 − 0.288i)9-s + (0.644 − 1.11i)11-s + 0.353·13-s − 1.10·15-s + (0.485 − 0.840i)17-s + (−0.146 − 0.253i)19-s + (−0.248 − 0.521i)21-s + (0.417 + 0.722i)23-s + (−1.32 + 2.29i)25-s − 0.192·27-s + 0.422·29-s + (−0.0898 + 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.928 + 0.371i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.928 + 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.610842989\)
\(L(\frac12)\) \(\approx\) \(1.610842989\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 2.17i)T \)
good5 \( 1 + (2.13 + 3.70i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.13 + 3.70i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.27T + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.637 + 1.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.27T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.63 - 4.56i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 7.27T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.862 + 1.49i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.13 - 5.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.63 - 6.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (1.63 - 2.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.77 + 3.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.274T + 83T^{2} \)
89 \( 1 + (2.27 + 3.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−8.877039293466155489369137507442, −8.519177123872209473969498383208, −7.74583428642945555546386105862, −7.09624258510186858245766596125, −5.82280641872155081152098719693, −4.90192741374399510967025873190, −4.10367944639262610177503391691, −3.25703942555577172315346299826, −1.32633112781309284140693355341, −0.73297124190971408099118612339, 2.01803034621738247170465210646, 2.98657356086379241540628531753, 3.89914073428811462065974805645, 4.63506943663467239109273843772, 6.00441758629331811877208673786, 6.70539274080299151270118362105, 7.67877645598971341132479086172, 8.186327328102716947615069774851, 9.202130319957285429580977101386, 10.05246654184771906210609957370

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