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Properties

Label 2-14-7.6-c4-0-3
Degree $2$
Conductor $14$
Sign $0.907 + 0.420i$
Analytic cond. $1.44717$
Root an. cond. $1.20298$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·2-s − 11.2i·3-s + 8.00·4-s + 43.1i·5-s − 31.8i·6-s + (−44.4 − 20.6i)7-s + 22.6·8-s − 46.0·9-s + 122. i·10-s − 11.8·11-s − 90.1i·12-s + 20.6i·13-s + (−125. − 58.2i)14-s + 486.·15-s + 64.0·16-s − 289. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.25i·3-s + 0.500·4-s + 1.72i·5-s − 0.885i·6-s + (−0.907 − 0.420i)7-s + 0.353·8-s − 0.568·9-s + 1.22i·10-s − 0.0977·11-s − 0.626i·12-s + 0.121i·13-s + (−0.641 − 0.297i)14-s + 2.16·15-s + 0.250·16-s − 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.907 + 0.420i$
Analytic conductor: \(1.44717\)
Root analytic conductor: \(1.20298\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :2),\ 0.907 + 0.420i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.49132 - 0.328848i\)
\(L(\frac12)\) \(\approx\) \(1.49132 - 0.328848i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82T \)
7 \( 1 + (44.4 + 20.6i)T \)
good3 \( 1 + 11.2iT - 81T^{2} \)
5 \( 1 - 43.1iT - 625T^{2} \)
11 \( 1 + 11.8T + 1.46e4T^{2} \)
13 \( 1 - 20.6iT - 2.85e4T^{2} \)
17 \( 1 + 289. iT - 8.35e4T^{2} \)
19 \( 1 + 104. iT - 1.30e5T^{2} \)
23 \( 1 - 73.5T + 2.79e5T^{2} \)
29 \( 1 - 950.T + 7.07e5T^{2} \)
31 \( 1 - 1.38e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.27e3T + 1.87e6T^{2} \)
41 \( 1 - 1.30e3iT - 2.82e6T^{2} \)
43 \( 1 + 96.2T + 3.41e6T^{2} \)
47 \( 1 - 186. iT - 4.87e6T^{2} \)
53 \( 1 - 4.37e3T + 7.89e6T^{2} \)
59 \( 1 - 1.65e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.20e3iT - 1.38e7T^{2} \)
67 \( 1 - 552.T + 2.01e7T^{2} \)
71 \( 1 + 8.48e3T + 2.54e7T^{2} \)
73 \( 1 - 317. iT - 2.83e7T^{2} \)
79 \( 1 + 624.T + 3.89e7T^{2} \)
83 \( 1 + 7.66e3iT - 4.74e7T^{2} \)
89 \( 1 + 4.19e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.29e4iT - 8.85e7T^{2} \)
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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

−18.85997381211740234959346877948, −17.83316130000728368102396877753, −15.82183953699284564450725827768, −14.25460658063141576353703847217, −13.40110562771398620534322710302, −11.93294322758369048840350546679, −10.39586744061276120199015969284, −7.21662346057119547636785436441, −6.54951202933580847798732327619, −2.92183054637341677873953587233, 4.10267712144115808830696036155, 5.55734086118024202494047993454, 8.764331441211042225167862507396, 10.10888711979164478437958395981, 12.18627026451762225964163807564, 13.23293435735272338433698154774, 15.24466414371385794236095435166, 16.10726557116231365861127015097, 16.94168896675500315848356208053, 19.51129218575465628034938292586

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