Properties

Label 2-14-7.3-c2-0-0
Degree $2$
Conductor $14$
Sign $0.947 - 0.318i$
Analytic cond. $0.381472$
Root an. cond. $0.617634$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)2-s + (−3.62 − 2.09i)3-s + (−0.999 + 1.73i)4-s + (2.74 − 1.58i)5-s − 5.91i·6-s + (−2.24 + 6.63i)7-s − 2.82·8-s + (4.24 + 7.34i)9-s + (3.87 + 2.23i)10-s + (6.62 − 11.4i)11-s + (7.24 − 4.18i)12-s + 5.49i·13-s + (−9.70 + 1.94i)14-s − 13.2·15-s + (−2.00 − 3.46i)16-s + (−11.7 − 6.77i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−1.20 − 0.696i)3-s + (−0.249 + 0.433i)4-s + (0.548 − 0.316i)5-s − 0.985i·6-s + (−0.320 + 0.947i)7-s − 0.353·8-s + (0.471 + 0.816i)9-s + (0.387 + 0.223i)10-s + (0.601 − 1.04i)11-s + (0.603 − 0.348i)12-s + 0.422i·13-s + (−0.693 + 0.138i)14-s − 0.882·15-s + (−0.125 − 0.216i)16-s + (−0.690 − 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(0.381472\)
Root analytic conductor: \(0.617634\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :1),\ 0.947 - 0.318i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.685400 + 0.112094i\)
\(L(\frac12)\) \(\approx\) \(0.685400 + 0.112094i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
7 \( 1 + (2.24 - 6.63i)T \)
good3 \( 1 + (3.62 + 2.09i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.74 + 1.58i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-6.62 + 11.4i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 5.49iT - 169T^{2} \)
17 \( 1 + (11.7 + 6.77i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (0.621 - 0.358i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-1.13 - 1.96i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 20.4T + 841T^{2} \)
31 \( 1 + (-21.3 - 12.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (32.4 + 56.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 21.0iT - 1.68e3T^{2} \)
43 \( 1 - 6.48T + 1.84e3T^{2} \)
47 \( 1 + (-41.3 + 23.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (11.0 - 19.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (72.5 + 41.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-57.3 + 33.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (46.3 - 80.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 48.4T + 5.04e3T^{2} \)
73 \( 1 + (-113. - 65.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-38.1 - 66.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 107. iT - 6.88e3T^{2} \)
89 \( 1 + (145. - 83.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 25.5iT - 9.40e3T^{2} \)
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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

−19.12785436954195583261576469186, −17.93144598955751673202225066245, −16.95823834259196337226855058641, −15.83087428406578860078218621466, −13.89178078181923680281174341622, −12.57998488392336310216795537851, −11.45939919926564139447325209898, −8.967667850166112942040191886460, −6.59018977696893865850250341187, −5.53199225414551369734500404159, 4.48579382903562604163237868559, 6.41286003038764665169598384020, 9.908535361913521390313667049692, 10.68917498998943417116922782835, 12.16506698103550209906162500864, 13.75515344716880117872314244192, 15.41667580609977017558094543434, 17.03975579311497346629829026250, 17.77935336065475932600477836503, 19.69417313839934162247600962104

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