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Properties

Label 2-1344-48.35-c1-0-27
Degree $2$
Conductor $1344$
Sign $-0.145 - 0.989i$
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  + (0.629 + 1.61i)3-s + (3.08 + 3.08i)5-s + 7-s + (−2.20 + 2.03i)9-s + (1.97 − 1.97i)11-s + (3.27 + 3.27i)13-s + (−3.03 + 6.92i)15-s − 2.57i·17-s + (1.18 − 1.18i)19-s + (0.629 + 1.61i)21-s − 1.21i·23-s + 14.0i·25-s + (−4.66 − 2.28i)27-s + (1.33 − 1.33i)29-s − 5.84i·31-s + ⋯
L(s)  = 1  + (0.363 + 0.931i)3-s + (1.38 + 1.38i)5-s + 0.377·7-s + (−0.735 + 0.677i)9-s + (0.596 − 0.596i)11-s + (0.908 + 0.908i)13-s + (−0.784 + 1.78i)15-s − 0.623i·17-s + (0.272 − 0.272i)19-s + (0.137 + 0.352i)21-s − 0.252i·23-s + 2.81i·25-s + (−0.898 − 0.439i)27-s + (0.248 − 0.248i)29-s − 1.05i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.145 - 0.989i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.145 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.654209123\)
\(L(\frac12)\) \(\approx\) \(2.654209123\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.629 - 1.61i)T \)
7 \( 1 - T \)
good5 \( 1 + (-3.08 - 3.08i)T + 5iT^{2} \)
11 \( 1 + (-1.97 + 1.97i)T - 11iT^{2} \)
13 \( 1 + (-3.27 - 3.27i)T + 13iT^{2} \)
17 \( 1 + 2.57iT - 17T^{2} \)
19 \( 1 + (-1.18 + 1.18i)T - 19iT^{2} \)
23 \( 1 + 1.21iT - 23T^{2} \)
29 \( 1 + (-1.33 + 1.33i)T - 29iT^{2} \)
31 \( 1 + 5.84iT - 31T^{2} \)
37 \( 1 + (-5.18 + 5.18i)T - 37iT^{2} \)
41 \( 1 - 0.688T + 41T^{2} \)
43 \( 1 + (0.649 + 0.649i)T + 43iT^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + (3.84 + 3.84i)T + 53iT^{2} \)
59 \( 1 + (1.99 - 1.99i)T - 59iT^{2} \)
61 \( 1 + (6.62 + 6.62i)T + 61iT^{2} \)
67 \( 1 + (4.50 - 4.50i)T - 67iT^{2} \)
71 \( 1 + 3.91iT - 71T^{2} \)
73 \( 1 + 9.83iT - 73T^{2} \)
79 \( 1 + 3.86iT - 79T^{2} \)
83 \( 1 + (2.99 + 2.99i)T + 83iT^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−9.577208584751502631929584291811, −9.424579189012061011249731034035, −8.434587266940340931160751210092, −7.31463029695815840860835497294, −6.31147137297859871490528333360, −5.86880613697247046267937483449, −4.72016431860889642714701881955, −3.62498398535439951228186039736, −2.79544546780104924507547751551, −1.81304800443670768893350786031, 1.26025778681049942137937966749, 1.58510228022098492920803706631, 2.96397392789885170329013350660, 4.37148044481749978444625671445, 5.41117951896808758773556751530, 6.02017420406080747387100206974, 6.81293803597224047880261715329, 8.151165770383126841719151448114, 8.409027599653084862908646127256, 9.311274522032298398267159359669

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