Properties

Label 2-14-7.6-c12-0-3
Degree $2$
Conductor $14$
Sign $0.609 - 0.792i$
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 − 321. i·3-s + 2.04e3·4-s + 1.71e4i·5-s − 1.45e4i·6-s + (−7.17e4 + 9.32e4i)7-s + 9.26e4·8-s + 4.27e5·9-s + 7.76e5i·10-s + 1.62e6·11-s − 6.59e5i·12-s + 4.55e6i·13-s + (−3.24e6 + 4.21e6i)14-s + 5.52e6·15-s + 4.19e6·16-s − 1.51e7i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.441i·3-s + 0.500·4-s + 1.09i·5-s − 0.312i·6-s + (−0.609 + 0.792i)7-s + 0.353·8-s + 0.805·9-s + 0.776i·10-s + 0.920·11-s − 0.220i·12-s + 0.944i·13-s + (−0.431 + 0.560i)14-s + 0.484·15-s + 0.250·16-s − 0.627i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.609 - 0.792i$
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: \((2,\ 14,\ (\ :6),\ 0.609 - 0.792i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.44164 + 1.20186i\)
\(L(\frac12)\) \(\approx\) \(2.44164 + 1.20186i\)
\(L(7)\) 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 \( 1 + (7.17e4 - 9.32e4i)T \)
good3 \( 1 + 321. iT - 5.31e5T^{2} \)
5 \( 1 - 1.71e4iT - 2.44e8T^{2} \)
11 \( 1 - 1.62e6T + 3.13e12T^{2} \)
13 \( 1 - 4.55e6iT - 2.32e13T^{2} \)
17 \( 1 + 1.51e7iT - 5.82e14T^{2} \)
19 \( 1 - 8.19e7iT - 2.21e15T^{2} \)
23 \( 1 - 8.57e7T + 2.19e16T^{2} \)
29 \( 1 + 9.60e8T + 3.53e17T^{2} \)
31 \( 1 + 1.52e9iT - 7.87e17T^{2} \)
37 \( 1 + 1.82e9T + 6.58e18T^{2} \)
41 \( 1 + 6.08e8iT - 2.25e19T^{2} \)
43 \( 1 - 3.59e9T + 3.99e19T^{2} \)
47 \( 1 - 6.65e9iT - 1.16e20T^{2} \)
53 \( 1 - 1.15e10T + 4.91e20T^{2} \)
59 \( 1 + 5.91e10iT - 1.77e21T^{2} \)
61 \( 1 - 2.23e10iT - 2.65e21T^{2} \)
67 \( 1 - 5.96e10T + 8.18e21T^{2} \)
71 \( 1 - 6.47e10T + 1.64e22T^{2} \)
73 \( 1 - 1.47e10iT - 2.29e22T^{2} \)
79 \( 1 - 2.34e11T + 5.90e22T^{2} \)
83 \( 1 - 1.33e11iT - 1.06e23T^{2} \)
89 \( 1 + 3.77e11iT - 2.46e23T^{2} \)
97 \( 1 + 1.35e12iT - 6.93e23T^{2} \)
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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

−16.58196173383143804666949863186, −15.15623853517042178879440722267, −14.10740014301781800166351159927, −12.62689791113851442442212165856, −11.44863381639546917916728646941, −9.641331934704074909038446432159, −7.19979189381558792044455765776, −6.13266089862179480180297905727, −3.73322676802597192768627014611, −2.00586014192355762546262101593, 1.02894467920572099816435599676, 3.70248593720916924072742691250, 4.98603328929898363400003726986, 6.98955925128644232555701830462, 9.138604422670192945154068881571, 10.68831694649711871444688778747, 12.57484601618913159534580214982, 13.36785189416276499792814932276, 15.15767884119955413033441735182, 16.27299839824023358690906482560

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