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Properties

Label 2-1456-13.4-c1-0-34
Degree $2$
Conductor $1456$
Sign $-0.0535 + 0.998i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + (1.67 − 2.89i)3-s + 1.56i·5-s + (0.866 − 0.5i)7-s + (−4.09 − 7.09i)9-s + (2.48 + 1.43i)11-s + (2.99 + 2.00i)13-s + (4.52 + 2.61i)15-s + (−1.11 − 1.93i)17-s + (6.26 − 3.61i)19-s − 3.34i·21-s + (0.833 − 1.44i)23-s + 2.55·25-s − 17.3·27-s + (−2.41 + 4.18i)29-s + 0.597i·31-s + ⋯
L(s)  = 1  + (0.965 − 1.67i)3-s + 0.699i·5-s + (0.327 − 0.188i)7-s + (−1.36 − 2.36i)9-s + (0.748 + 0.432i)11-s + (0.830 + 0.556i)13-s + (1.16 + 0.675i)15-s + (−0.270 − 0.468i)17-s + (1.43 − 0.830i)19-s − 0.729i·21-s + (0.173 − 0.301i)23-s + 0.511·25-s − 3.33·27-s + (−0.448 + 0.776i)29-s + 0.107i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0535 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0535 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.0535 + 0.998i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -0.0535 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.579204206\)
\(L(\frac12)\) \(\approx\) \(2.579204206\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-2.99 - 2.00i)T \)
good3 \( 1 + (-1.67 + 2.89i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.56iT - 5T^{2} \)
11 \( 1 + (-2.48 - 1.43i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.11 + 1.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.26 + 3.61i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.833 + 1.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.41 - 4.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.597iT - 31T^{2} \)
37 \( 1 + (-0.0333 - 0.0192i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.88 + 3.97i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.04 + 8.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.02iT - 47T^{2} \)
53 \( 1 + 5.98T + 53T^{2} \)
59 \( 1 + (0.776 - 0.448i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.12 - 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.42 - 0.820i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.98 + 1.14i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 + 4.26T + 79T^{2} \)
83 \( 1 - 4.94iT - 83T^{2} \)
89 \( 1 + (-2.09 - 1.21i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.23 - 2.44i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.878099301888669910903306936223, −8.639727173706220537245886466413, −7.40340699251358038772441330769, −6.98794813913106821778150390913, −6.58511491422727825531454551914, −5.31907199288670394095384798140, −3.77880801312734387834508996169, −3.00658435546126890344297493588, −1.98419553210974642311107765384, −1.04618727321443967916706903238, 1.53918538524558811788497949262, 3.11717678558021086735280216149, 3.65165685093237682619595232638, 4.60855688580581747630509895232, 5.29054505947238059501731705811, 6.16901735018368902500593934765, 7.931614482985214648203282034655, 8.212523316321139310002478820084, 9.084936393026356918240273053417, 9.526343749011974945182523424660

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