Properties

Label 2-14-7.6-c12-0-1
Degree $2$
Conductor $14$
Sign $-0.629 - 0.777i$
Analytic cond. $12.7959$
Root an. cond. $3.57713$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 45.2·2-s + 1.07e3i·3-s + 2.04e3·4-s + 1.14e3i·5-s + 4.85e4i·6-s + (7.40e4 + 9.14e4i)7-s + 9.26e4·8-s − 6.17e5·9-s + 5.16e4i·10-s − 2.90e6·11-s + 2.19e6i·12-s − 4.91e5i·13-s + (3.35e6 + 4.13e6i)14-s − 1.22e6·15-s + 4.19e6·16-s + 3.15e7i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.47i·3-s + 0.500·4-s + 0.0730i·5-s + 1.03i·6-s + (0.629 + 0.777i)7-s + 0.353·8-s − 1.16·9-s + 0.0516i·10-s − 1.64·11-s + 0.735i·12-s − 0.101i·13-s + (0.444 + 0.549i)14-s − 0.107·15-s + 0.250·16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.629 - 0.777i$
Analytic conductor: \(12.7959\)
Root analytic conductor: \(3.57713\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :6),\ -0.629 - 0.777i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.07813 + 2.26034i\)
\(L(\frac12)\) \(\approx\) \(1.07813 + 2.26034i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 45.2T \)
7 \( 1 + (-7.40e4 - 9.14e4i)T \)
good3 \( 1 - 1.07e3iT - 5.31e5T^{2} \)
5 \( 1 - 1.14e3iT - 2.44e8T^{2} \)
11 \( 1 + 2.90e6T + 3.13e12T^{2} \)
13 \( 1 + 4.91e5iT - 2.32e13T^{2} \)
17 \( 1 - 3.15e7iT - 5.82e14T^{2} \)
19 \( 1 + 1.15e7iT - 2.21e15T^{2} \)
23 \( 1 - 6.83e6T + 2.19e16T^{2} \)
29 \( 1 - 5.16e8T + 3.53e17T^{2} \)
31 \( 1 + 1.26e9iT - 7.87e17T^{2} \)
37 \( 1 - 3.49e9T + 6.58e18T^{2} \)
41 \( 1 - 8.38e9iT - 2.25e19T^{2} \)
43 \( 1 + 6.25e8T + 3.99e19T^{2} \)
47 \( 1 + 1.58e10iT - 1.16e20T^{2} \)
53 \( 1 - 2.53e10T + 4.91e20T^{2} \)
59 \( 1 - 4.34e10iT - 1.77e21T^{2} \)
61 \( 1 + 7.58e9iT - 2.65e21T^{2} \)
67 \( 1 - 2.50e10T + 8.18e21T^{2} \)
71 \( 1 - 1.41e11T + 1.64e22T^{2} \)
73 \( 1 - 1.37e11iT - 2.29e22T^{2} \)
79 \( 1 + 1.68e11T + 5.90e22T^{2} \)
83 \( 1 + 7.26e10iT - 1.06e23T^{2} \)
89 \( 1 + 5.99e11iT - 2.46e23T^{2} \)
97 \( 1 + 5.96e11iT - 6.93e23T^{2} \)
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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

−16.61315533705551098742063626581, −15.37522719785454593331694174437, −14.86938654358523746692939506824, −12.98628380009316534311317160202, −11.22664977072879585958586466992, −10.15242705258501014476588602296, −8.291652850776656815445296461354, −5.60976235784417813210840043704, −4.48562602023509759167347744055, −2.70764637171800086815188574881, 0.875809490556841484550676879137, 2.56032029814957121773463574944, 5.06375493672508534352450625753, 6.95390264860430101577309111477, 7.952504165606032303403284459038, 10.76778983476606405552839151289, 12.26003627626946427299681939041, 13.34494859274571061498715915762, 14.20720451115603135582696542276, 16.06565530656073541907367629991

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