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Properties

Label 2-14-7.2-c5-0-3
Degree $2$
Conductor $14$
Sign $-0.159 + 0.987i$
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 + (−7.90 − 13.6i)3-s + (−7.99 − 13.8i)4-s + (10.5 − 18.1i)5-s − 63.2·6-s + (126. + 28.7i)7-s − 63.9·8-s + (−3.37 + 5.84i)9-s + (−42 − 72.7i)10-s + (312. + 542. i)11-s + (−126. + 218. i)12-s − 206.·13-s + (352. − 380. i)14-s − 331.·15-s + (−128 + 221. i)16-s + (−530. − 919. i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.506 − 0.877i)3-s + (−0.249 − 0.433i)4-s + (0.187 − 0.325i)5-s − 0.716·6-s + (0.975 + 0.221i)7-s − 0.353·8-s + (−0.0138 + 0.0240i)9-s + (−0.132 − 0.230i)10-s + (0.779 + 1.35i)11-s + (−0.253 + 0.438i)12-s − 0.339·13-s + (0.480 − 0.518i)14-s − 0.380·15-s + (−0.125 + 0.216i)16-s + (−0.445 − 0.771i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.159 + 0.987i$
Analytic conductor: \(2.24537\)
Root analytic conductor: \(1.49845\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :5/2),\ -0.159 + 0.987i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.899201 - 1.05633i\)
\(L(\frac12)\) \(\approx\) \(0.899201 - 1.05633i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-126. - 28.7i)T \)
good3 \( 1 + (7.90 + 13.6i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-10.5 + 18.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-312. - 542. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 206.T + 3.71e5T^{2} \)
17 \( 1 + (530. + 919. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-941. + 1.63e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.85e3 - 3.21e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 123.T + 2.05e7T^{2} \)
31 \( 1 + (-4.55e3 - 7.88e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-3.01e3 + 5.22e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.72e4T + 1.15e8T^{2} \)
43 \( 1 - 5.40e3T + 1.47e8T^{2} \)
47 \( 1 + (937. - 1.62e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (9.35e3 + 1.62e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.26e3 - 2.19e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.04e3 + 1.81e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.93e4 + 5.07e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 3.12e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.57e3 - 6.19e3i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.48e3 - 2.58e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 4.59e4T + 3.93e9T^{2} \)
89 \( 1 + (4.95e4 - 8.57e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.15e5T + 8.58e9T^{2} \)
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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

−17.97633350844609795866525274326, −17.53597618641164270681238165417, −15.22279511680101025500661239458, −13.73182614671858934765727008872, −12.31387751233513107520349799710, −11.52486750802856793276955225786, −9.417659084848266015365753688325, −7.09934390351978579673289824660, −4.96468239122021928966595986271, −1.55926070621377952739431478062, 4.29660460345777588684773726639, 6.02849997125945547044897602226, 8.264826062189071139442519716551, 10.34284283347555789522697803753, 11.67694431780276305059640724738, 13.84252505886041243411572135546, 14.87581373175588168750187228589, 16.36337917563931949964265670347, 17.13803605614795906145051068394, 18.66977945250395786486196673308

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