Properties

Label 2-14-7.3-c6-0-0
Degree $2$
Conductor $14$
Sign $-0.974 + 0.224i$
Analytic cond. $3.22075$
Root an. cond. $1.79464$
Motivic weight $6$
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 + 4.89i)2-s + (−36.7 − 21.2i)3-s + (−15.9 + 27.7i)4-s + (−162. + 93.7i)5-s − 239. i·6-s + (141. − 312. i)7-s − 181.·8-s + (535. + 927. i)9-s + (−918. − 530. i)10-s + (−555. + 962. i)11-s + (1.17e3 − 678. i)12-s + 706. i·13-s + (1.93e3 − 192. i)14-s + 7.95e3·15-s + (−512. − 886. i)16-s + (−7.33e3 − 4.23e3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−1.36 − 0.785i)3-s + (−0.249 + 0.433i)4-s + (−1.29 + 0.749i)5-s − 1.11i·6-s + (0.411 − 0.911i)7-s − 0.353·8-s + (0.734 + 1.27i)9-s + (−0.918 − 0.530i)10-s + (−0.417 + 0.723i)11-s + (0.680 − 0.392i)12-s + 0.321i·13-s + (0.703 − 0.0700i)14-s + 2.35·15-s + (−0.125 − 0.216i)16-s + (−1.49 − 0.861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.974 + 0.224i$
Analytic conductor: \(3.22075\)
Root analytic conductor: \(1.79464\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :3),\ -0.974 + 0.224i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.00921571 - 0.0812319i\)
\(L(\frac12)\) \(\approx\) \(0.00921571 - 0.0812319i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.82 - 4.89i)T \)
7 \( 1 + (-141. + 312. i)T \)
good3 \( 1 + (36.7 + 21.2i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (162. - 93.7i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (555. - 962. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 706. iT - 4.82e6T^{2} \)
17 \( 1 + (7.33e3 + 4.23e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-22.0 + 12.7i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-424. - 735. i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 1.51e4T + 5.94e8T^{2} \)
31 \( 1 + (1.20e3 + 696. i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-4.64e3 - 8.04e3i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 1.09e5iT - 4.75e9T^{2} \)
43 \( 1 + 4.55e4T + 6.32e9T^{2} \)
47 \( 1 + (-1.19e5 + 6.92e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (3.09e4 - 5.35e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-6.80e4 - 3.92e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.51e5 - 8.73e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-2.17e5 + 3.77e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 5.61e5T + 1.28e11T^{2} \)
73 \( 1 + (1.92e5 + 1.10e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (3.33e5 + 5.77e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 6.53e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.14e5 + 6.61e4i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 1.59e6iT - 8.32e11T^{2} \)
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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

−18.52586266136478741265667057117, −17.63827577863054274051413971691, −16.37562777080321969994998800084, −15.09860309205796285495790031491, −13.35317877876274823530668912210, −11.83950897215917063279934054986, −10.95286841991577160020943939037, −7.55035587508063884003396003026, −6.79581831359207248185149357452, −4.56593239874600100182625789833, 0.06301963162225269464454508655, 4.26718284075360051888313407319, 5.58322959317473051261828330436, 8.676689252786952678461792689477, 10.83530825046464348688806416065, 11.63140187449323091173064775614, 12.70830630250047383097237723385, 15.30135836739645629999603408427, 15.99465338923401800473278822160, 17.47885838666297571642242453019

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