Properties

Label 2-14-7.3-c14-0-4
Degree $2$
Conductor $14$
Sign $-0.126 + 0.991i$
Analytic cond. $17.4060$
Root an. cond. $4.17205$
Motivic weight $14$
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 − 78.3i)2-s + (−3.57e3 − 2.06e3i)3-s + (−4.09e3 + 7.09e3i)4-s + (8.10e4 − 4.67e4i)5-s + 3.73e5i·6-s + (5.87e5 + 5.77e5i)7-s + 7.41e5·8-s + (6.11e6 + 1.05e7i)9-s + (−7.33e6 − 4.23e6i)10-s + (1.07e7 − 1.85e7i)11-s + (2.92e7 − 1.68e7i)12-s + 9.80e6i·13-s + (1.86e7 − 7.21e7i)14-s − 3.85e8·15-s + (−3.35e7 − 5.81e7i)16-s + (5.15e8 + 2.97e8i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−1.63 − 0.942i)3-s + (−0.249 + 0.433i)4-s + (1.03 − 0.599i)5-s + 1.33i·6-s + (0.712 + 0.701i)7-s + 0.353·8-s + (1.27 + 2.21i)9-s + (−0.733 − 0.423i)10-s + (0.550 − 0.952i)11-s + (0.816 − 0.471i)12-s + 0.156i·13-s + (0.177 − 0.684i)14-s − 2.25·15-s + (−0.125 − 0.216i)16-s + (1.25 + 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.126 + 0.991i$
Analytic conductor: \(17.4060\)
Root analytic conductor: \(4.17205\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :7),\ -0.126 + 0.991i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.791462 - 0.898781i\)
\(L(\frac12)\) \(\approx\) \(0.791462 - 0.898781i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (45.2 + 78.3i)T \)
7 \( 1 + (-5.87e5 - 5.77e5i)T \)
good3 \( 1 + (3.57e3 + 2.06e3i)T + (2.39e6 + 4.14e6i)T^{2} \)
5 \( 1 + (-8.10e4 + 4.67e4i)T + (3.05e9 - 5.28e9i)T^{2} \)
11 \( 1 + (-1.07e7 + 1.85e7i)T + (-1.89e14 - 3.28e14i)T^{2} \)
13 \( 1 - 9.80e6iT - 3.93e15T^{2} \)
17 \( 1 + (-5.15e8 - 2.97e8i)T + (8.41e16 + 1.45e17i)T^{2} \)
19 \( 1 + (1.96e8 - 1.13e8i)T + (3.99e17 - 6.91e17i)T^{2} \)
23 \( 1 + (7.21e8 + 1.24e9i)T + (-5.79e18 + 1.00e19i)T^{2} \)
29 \( 1 - 1.54e10T + 2.97e20T^{2} \)
31 \( 1 + (-2.91e10 - 1.68e10i)T + (3.78e20 + 6.55e20i)T^{2} \)
37 \( 1 + (1.75e10 + 3.03e10i)T + (-4.50e21 + 7.80e21i)T^{2} \)
41 \( 1 - 9.80e10iT - 3.79e22T^{2} \)
43 \( 1 + 3.91e11T + 7.38e22T^{2} \)
47 \( 1 + (-6.95e11 + 4.01e11i)T + (1.28e23 - 2.22e23i)T^{2} \)
53 \( 1 + (-9.24e11 + 1.60e12i)T + (-6.89e23 - 1.19e24i)T^{2} \)
59 \( 1 + (9.16e11 + 5.29e11i)T + (3.09e24 + 5.36e24i)T^{2} \)
61 \( 1 + (5.05e11 - 2.91e11i)T + (4.93e24 - 8.55e24i)T^{2} \)
67 \( 1 + (-5.19e12 + 8.99e12i)T + (-1.83e25 - 3.18e25i)T^{2} \)
71 \( 1 + 5.47e12T + 8.27e25T^{2} \)
73 \( 1 + (-2.19e12 - 1.26e12i)T + (6.10e25 + 1.05e26i)T^{2} \)
79 \( 1 + (3.31e12 + 5.73e12i)T + (-1.84e26 + 3.19e26i)T^{2} \)
83 \( 1 + 1.05e13iT - 7.36e26T^{2} \)
89 \( 1 + (2.07e13 - 1.19e13i)T + (9.78e26 - 1.69e27i)T^{2} \)
97 \( 1 - 1.03e14iT - 6.52e27T^{2} \)
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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

−16.57625691775763727583684090543, −13.79626702280993023891255918811, −12.47216218911066502102321705154, −11.67161349452280729596333694880, −10.30753867745080989500312805769, −8.361092629351579040593172536351, −6.21952782274343982914919122660, −5.16655865906593087487633082006, −1.78966370199260471053543189647, −0.899263361815036445163496445790, 1.04165492530931547247547726871, 4.51189841116919996370008935947, 5.75639772158882780924640483180, 7.00557467586186044500717170325, 9.790069348465159731480898991873, 10.39314251533131651804017838619, 11.85819562260960303212050163730, 14.12014775464293727998000671508, 15.38554751931883828887043551150, 16.88148905181978476425529613027

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