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Properties

Label 2-14-7.4-c5-0-0
Degree $2$
Conductor $14$
Sign $-0.219 - 0.975i$
Analytic cond. $2.24537$
Root an. cond. $1.49845$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 3.46i)2-s + (−12.3 + 21.4i)3-s + (−7.99 + 13.8i)4-s + (18.0 + 31.2i)5-s + 99.1·6-s + (−124. + 36.3i)7-s + 63.9·8-s + (−185. − 321. i)9-s + (72.2 − 125. i)10-s + (−77.7 + 134. i)11-s + (−198. − 343. i)12-s + 1.15e3·13-s + (374. + 358. i)14-s − 894.·15-s + (−128 − 221. i)16-s + (−619. + 1.07e3i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.794 + 1.37i)3-s + (−0.249 + 0.433i)4-s + (0.322 + 0.559i)5-s + 1.12·6-s + (−0.959 + 0.280i)7-s + 0.353·8-s + (−0.763 − 1.32i)9-s + (0.228 − 0.395i)10-s + (−0.193 + 0.335i)11-s + (−0.397 − 0.688i)12-s + 1.90·13-s + (0.511 + 0.488i)14-s − 1.02·15-s + (−0.125 − 0.216i)16-s + (−0.519 + 0.899i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.219 - 0.975i$
Analytic conductor: \(2.24537\)
Root analytic conductor: \(1.49845\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :5/2),\ -0.219 - 0.975i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.439915 + 0.549721i\)
\(L(\frac12)\) \(\approx\) \(0.439915 + 0.549721i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 3.46i)T \)
7 \( 1 + (124. - 36.3i)T \)
good3 \( 1 + (12.3 - 21.4i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-18.0 - 31.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (77.7 - 134. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 1.15e3T + 3.71e5T^{2} \)
17 \( 1 + (619. - 1.07e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-140. - 242. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.74e3 - 3.01e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 5.65e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.15e3 + 2.00e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-1.16e3 - 2.02e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 3.81e3T + 1.15e8T^{2} \)
43 \( 1 - 3.92e3T + 1.47e8T^{2} \)
47 \( 1 + (-5.55e3 - 9.62e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-5.59e3 + 9.68e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-3.00e3 + 5.20e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-7.41e3 - 1.28e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.14e4 + 3.72e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.99e4T + 1.80e9T^{2} \)
73 \( 1 + (2.27e4 - 3.94e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (5.44e4 + 9.43e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 5.58e4T + 3.93e9T^{2} \)
89 \( 1 + (-4.77e4 - 8.27e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.50e4T + 8.58e9T^{2} \)
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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

−18.86699168672612883433820243918, −17.61018684443540983824543082133, −16.29166892400824698199034599683, −15.31118133356509020934879327303, −13.15897642463810621901322979567, −11.27336978703424235051717979882, −10.37212221531966414093657590799, −9.163537658794135789585449238631, −6.02656161643420493817524630757, −3.68890858346405106676408285596, 0.792594183323742904786243806892, 5.84453914299306711760414322119, 6.99171939447637807262730378205, 8.853431617744349594144486207425, 11.04842716936764132080669719118, 12.91221065039988757568708160375, 13.56253137406316654846564392051, 16.02352740532056570693387484103, 16.92033024082897982214896710613, 18.22553083833260898132662616310

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