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Properties

Label 2-1344-48.35-c1-0-8
Degree $2$
Conductor $1344$
Sign $-0.989 - 0.141i$
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  + (−1.18 + 1.26i)3-s + (2.84 + 2.84i)5-s + 7-s + (−0.196 − 2.99i)9-s + (−3.31 + 3.31i)11-s + (−2.31 − 2.31i)13-s + (−6.95 + 0.228i)15-s + 0.814i·17-s + (−4.39 + 4.39i)19-s + (−1.18 + 1.26i)21-s − 3.30i·23-s + 11.1i·25-s + (4.01 + 3.29i)27-s + (−3.25 + 3.25i)29-s − 3.89i·31-s + ⋯
L(s)  = 1  + (−0.683 + 0.729i)3-s + (1.27 + 1.27i)5-s + 0.377·7-s + (−0.0656 − 0.997i)9-s + (−1.00 + 1.00i)11-s + (−0.641 − 0.641i)13-s + (−1.79 + 0.0590i)15-s + 0.197i·17-s + (−1.00 + 1.00i)19-s + (−0.258 + 0.275i)21-s − 0.688i·23-s + 2.23i·25-s + (0.773 + 0.634i)27-s + (−0.604 + 0.604i)29-s − 0.699i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.989 - 0.141i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.989 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.042260743\)
\(L(\frac12)\) \(\approx\) \(1.042260743\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 - 1.26i)T \)
7 \( 1 - T \)
good5 \( 1 + (-2.84 - 2.84i)T + 5iT^{2} \)
11 \( 1 + (3.31 - 3.31i)T - 11iT^{2} \)
13 \( 1 + (2.31 + 2.31i)T + 13iT^{2} \)
17 \( 1 - 0.814iT - 17T^{2} \)
19 \( 1 + (4.39 - 4.39i)T - 19iT^{2} \)
23 \( 1 + 3.30iT - 23T^{2} \)
29 \( 1 + (3.25 - 3.25i)T - 29iT^{2} \)
31 \( 1 + 3.89iT - 31T^{2} \)
37 \( 1 + (3.65 - 3.65i)T - 37iT^{2} \)
41 \( 1 + 3.20T + 41T^{2} \)
43 \( 1 + (1.63 + 1.63i)T + 43iT^{2} \)
47 \( 1 - 6.70T + 47T^{2} \)
53 \( 1 + (-7.69 - 7.69i)T + 53iT^{2} \)
59 \( 1 + (-7.31 + 7.31i)T - 59iT^{2} \)
61 \( 1 + (5.54 + 5.54i)T + 61iT^{2} \)
67 \( 1 + (-0.0553 + 0.0553i)T - 67iT^{2} \)
71 \( 1 - 6.50iT - 71T^{2} \)
73 \( 1 - 6.05iT - 73T^{2} \)
79 \( 1 - 15.1iT - 79T^{2} \)
83 \( 1 + (4.25 + 4.25i)T + 83iT^{2} \)
89 \( 1 + 2.31T + 89T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−10.14664553037154385382926739389, −9.658636303501332649263209478289, −8.449438166799767145641468775875, −7.32569644523441318639951787263, −6.62926677894489520218180572118, −5.70288933518248685062361039447, −5.21255545236610086039281450459, −4.08237602182545708964502438379, −2.82328416074216682214517562614, −1.96364862530166408203186456379, 0.43256405985367058684759160949, 1.71597571666438687715073947749, 2.50643541378342074218257024495, 4.47614058498291642413342268084, 5.28297913704671766492998598888, 5.66660895221850928986675640845, 6.63812867798995362727048000431, 7.58212338717942072302849504100, 8.564385954183832262654009256257, 9.031376521230051039236802249968

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