Properties

Label 2-14-7.2-c15-0-4
Degree $2$
Conductor $14$
Sign $0.999 - 0.00920i$
Analytic cond. $19.9770$
Root an. cond. $4.46957$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (64 − 110. i)2-s + (350. + 607. i)3-s + (−8.19e3 − 1.41e4i)4-s + (−6.45e3 + 1.11e4i)5-s + 8.97e4·6-s + (−1.17e5 + 2.17e6i)7-s − 2.09e6·8-s + (6.92e6 − 1.20e7i)9-s + (8.25e5 + 1.43e6i)10-s + (4.09e7 + 7.08e7i)11-s + (5.74e6 − 9.94e6i)12-s − 3.93e7·13-s + (2.33e8 + 1.52e8i)14-s − 9.04e6·15-s + (−1.34e8 + 2.32e8i)16-s + (1.47e9 + 2.55e9i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.0925 + 0.160i)3-s + (−0.249 − 0.433i)4-s + (−0.0369 + 0.0639i)5-s + 0.130·6-s + (−0.0541 + 0.998i)7-s − 0.353·8-s + (0.482 − 0.836i)9-s + (0.0261 + 0.0452i)10-s + (0.633 + 1.09i)11-s + (0.0462 − 0.0801i)12-s − 0.173·13-s + (0.592 + 0.386i)14-s − 0.0136·15-s + (−0.125 + 0.216i)16-s + (0.873 + 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00920i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.999 - 0.00920i$
Analytic conductor: \(19.9770\)
Root analytic conductor: \(4.46957\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :15/2),\ 0.999 - 0.00920i)\)

Particular Values

\(L(8)\) \(\approx\) \(2.41747 + 0.0111324i\)
\(L(\frac12)\) \(\approx\) \(2.41747 + 0.0111324i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-64 + 110. i)T \)
7 \( 1 + (1.17e5 - 2.17e6i)T \)
good3 \( 1 + (-350. - 607. i)T + (-7.17e6 + 1.24e7i)T^{2} \)
5 \( 1 + (6.45e3 - 1.11e4i)T + (-1.52e10 - 2.64e10i)T^{2} \)
11 \( 1 + (-4.09e7 - 7.08e7i)T + (-2.08e15 + 3.61e15i)T^{2} \)
13 \( 1 + 3.93e7T + 5.11e16T^{2} \)
17 \( 1 + (-1.47e9 - 2.55e9i)T + (-1.43e18 + 2.47e18i)T^{2} \)
19 \( 1 + (-2.67e9 + 4.62e9i)T + (-7.59e18 - 1.31e19i)T^{2} \)
23 \( 1 + (-8.95e9 + 1.55e10i)T + (-1.33e20 - 2.30e20i)T^{2} \)
29 \( 1 - 1.47e11T + 8.62e21T^{2} \)
31 \( 1 + (-2.97e10 - 5.15e10i)T + (-1.17e22 + 2.03e22i)T^{2} \)
37 \( 1 + (1.06e11 - 1.83e11i)T + (-1.66e23 - 2.88e23i)T^{2} \)
41 \( 1 + 1.79e12T + 1.55e24T^{2} \)
43 \( 1 + 9.09e11T + 3.17e24T^{2} \)
47 \( 1 + (2.19e12 - 3.79e12i)T + (-6.03e24 - 1.04e25i)T^{2} \)
53 \( 1 + (-4.71e12 - 8.16e12i)T + (-3.65e25 + 6.33e25i)T^{2} \)
59 \( 1 + (3.85e11 + 6.67e11i)T + (-1.82e26 + 3.16e26i)T^{2} \)
61 \( 1 + (8.40e11 - 1.45e12i)T + (-3.01e26 - 5.21e26i)T^{2} \)
67 \( 1 + (9.27e12 + 1.60e13i)T + (-1.23e27 + 2.13e27i)T^{2} \)
71 \( 1 + 1.51e12T + 5.87e27T^{2} \)
73 \( 1 + (-4.49e13 - 7.79e13i)T + (-4.45e27 + 7.71e27i)T^{2} \)
79 \( 1 + (-1.22e14 + 2.12e14i)T + (-1.45e28 - 2.52e28i)T^{2} \)
83 \( 1 + 3.41e14T + 6.11e28T^{2} \)
89 \( 1 + (-3.29e14 + 5.69e14i)T + (-8.70e28 - 1.50e29i)T^{2} \)
97 \( 1 - 7.75e14T + 6.33e29T^{2} \)
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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

−15.39975760189733692062227582235, −14.66823556558400922509126522333, −12.72299306025036286374562742389, −11.93418788335363413080236793890, −10.11750973222001138493062755702, −8.911018699223210769142601128243, −6.59167110550634550355968994325, −4.75503424976391476136648126880, −3.09868898518831409919074705540, −1.36369035111207790756725135411, 0.936763296339814996156061323593, 3.41042542861194020520185272405, 5.07540826210314236005145808258, 6.90543836600314616196688832702, 8.108925114329045290843179506467, 10.04682269778252416164719159941, 11.81699509011313851978171604999, 13.57668596861283734836671457374, 14.16606076180372459821310686520, 16.14360374690172346619829489077

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