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Properties

Label 2-14-7.5-c4-0-1
Degree $2$
Conductor $14$
Sign $0.701 + 0.712i$
Analytic cond. $1.44717$
Root an. cond. $1.20298$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 2.44i)2-s + (3.98 − 2.30i)3-s + (−3.99 − 6.92i)4-s + (5.01 + 2.89i)5-s − 13.0i·6-s + (−7 + 48.4i)7-s − 22.6·8-s + (−29.9 + 51.8i)9-s + (14.1 − 8.18i)10-s + (−5.01 − 8.68i)11-s + (−31.8 − 18.4i)12-s − 190. i·13-s + (108. + 85.7i)14-s + 26.6·15-s + (−32.0 + 55.4i)16-s + (365. − 210. i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.442 − 0.255i)3-s + (−0.249 − 0.433i)4-s + (0.200 + 0.115i)5-s − 0.361i·6-s + (−0.142 + 0.989i)7-s − 0.353·8-s + (−0.369 + 0.639i)9-s + (0.141 − 0.0818i)10-s + (−0.0414 − 0.0717i)11-s + (−0.221 − 0.127i)12-s − 1.12i·13-s + (0.555 + 0.437i)14-s + 0.118·15-s + (−0.125 + 0.216i)16-s + (1.26 − 0.729i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(1.44717\)
Root analytic conductor: \(1.20298\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :2),\ 0.701 + 0.712i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.31407 - 0.550707i\)
\(L(\frac12)\) \(\approx\) \(1.31407 - 0.550707i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 2.44i)T \)
7 \( 1 + (7 - 48.4i)T \)
good3 \( 1 + (-3.98 + 2.30i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-5.01 - 2.89i)T + (312.5 + 541. i)T^{2} \)
11 \( 1 + (5.01 + 8.68i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + 190. iT - 2.85e4T^{2} \)
17 \( 1 + (-365. + 210. i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (374. + 216. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (460. - 798. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 877.T + 7.07e5T^{2} \)
31 \( 1 + (-627. + 362. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-270. + 468. i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 894. iT - 2.82e6T^{2} \)
43 \( 1 + 1.24e3T + 3.41e6T^{2} \)
47 \( 1 + (-1.51e3 - 875. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (406. + 703. i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-2.47e3 + 1.42e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-5.05e3 - 2.91e3i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-1.10e3 - 1.90e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 3.40e3T + 2.54e7T^{2} \)
73 \( 1 + (8.13e3 - 4.69e3i)T + (1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (1.17e3 - 2.03e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 3.75e3iT - 4.74e7T^{2} \)
89 \( 1 + (5.51e3 + 3.18e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 6.37e3iT - 8.85e7T^{2} \)
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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

−18.99900759558439714506484042535, −17.73074790526282982126253626749, −15.73867699305448813221643679268, −14.34334522845697290669127498135, −13.11187538038583761576503023250, −11.73437019802497804966959713482, −9.993828081510207705370444542681, −8.223898496734716742371308947196, −5.58140145221954869577744944150, −2.68841586638820840173826987282, 4.00420387118227097261736069547, 6.45202168228227949375328516930, 8.364604495606938594391378764805, 10.07593056200087511253200364713, 12.26812170209811569708195495775, 13.91526165140878459477615966428, 14.73967875947547466816567737754, 16.40776166626998603385547265176, 17.31790852910112689658981491853, 19.08666561849137382797741618290

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