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Properties

Label 2-1344-56.37-c1-0-26
Degree $2$
Conductor $1344$
Sign $-0.378 + 0.925i$
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  + (0.866 − 0.5i)3-s + (−0.866 − 0.5i)5-s + (0.5 − 2.59i)7-s + (0.499 − 0.866i)9-s + (−0.866 + 0.5i)11-s − 2i·13-s − 0.999·15-s + (2 + 3.46i)17-s + (−0.866 − 2.5i)21-s + (3 − 5.19i)23-s + (−2 − 3.46i)25-s − 0.999i·27-s + i·29-s + (−2.5 − 4.33i)31-s + (−0.499 + 0.866i)33-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (−0.387 − 0.223i)5-s + (0.188 − 0.981i)7-s + (0.166 − 0.288i)9-s + (−0.261 + 0.150i)11-s − 0.554i·13-s − 0.258·15-s + (0.485 + 0.840i)17-s + (−0.188 − 0.545i)21-s + (0.625 − 1.08i)23-s + (−0.400 − 0.692i)25-s − 0.192i·27-s + 0.185i·29-s + (−0.449 − 0.777i)31-s + (−0.0870 + 0.150i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.378 + 0.925i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.378 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.544875251\)
\(L(\frac12)\) \(\approx\) \(1.544875251\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (0.866 + 0.5i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - iT - 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.79 - 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.92 + 4i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 + i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15iT - 83T^{2} \)
89 \( 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 11T + 97T^{2} \)
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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

−9.277449691453581434514368438637, −8.336961756644971806525029220146, −7.84241205779589319866633664550, −7.09744299186581497780181280717, −6.17227403806459947932918502970, −5.00598515301182803404182046217, −4.08295761939306965428945989026, −3.28854451877169357426320066698, −1.96209583144523225908970881028, −0.59563149061731777618975709970, 1.69547073000533493315652736021, 2.90630534786399352740371364961, 3.61053121792848306667144081156, 4.91168785790800944112264385726, 5.49177730908438182867842717519, 6.71053585019791265093193051805, 7.55367232529056273835777742948, 8.271913450239107592901785450955, 9.200823050942744737972382598391, 9.544033309782480611246250136607

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