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Properties

Label 2-14-7.6-c8-0-2
Degree $2$
Conductor $14$
Sign $0.286 + 0.958i$
Analytic cond. $5.70330$
Root an. cond. $2.38815$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11.3·2-s − 63.0i·3-s + 128.·4-s − 390. i·5-s − 713. i·6-s + (−687. − 2.30e3i)7-s + 1.44e3·8-s + 2.58e3·9-s − 4.41e3i·10-s + 4.00e3·11-s − 8.07e3i·12-s − 350. i·13-s + (−7.77e3 − 2.60e4i)14-s − 2.46e4·15-s + 1.63e4·16-s + 9.70e4i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.778i·3-s + 0.500·4-s − 0.624i·5-s − 0.550i·6-s + (−0.286 − 0.958i)7-s + 0.353·8-s + 0.393·9-s − 0.441i·10-s + 0.273·11-s − 0.389i·12-s − 0.0122i·13-s + (−0.202 − 0.677i)14-s − 0.486·15-s + 0.250·16-s + 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(5.70330\)
Root analytic conductor: \(2.38815\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :4),\ 0.286 + 0.958i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.86219 - 1.38709i\)
\(L(\frac12)\) \(\approx\) \(1.86219 - 1.38709i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 11.3T \)
7 \( 1 + (687. + 2.30e3i)T \)
good3 \( 1 + 63.0iT - 6.56e3T^{2} \)
5 \( 1 + 390. iT - 3.90e5T^{2} \)
11 \( 1 - 4.00e3T + 2.14e8T^{2} \)
13 \( 1 + 350. iT - 8.15e8T^{2} \)
17 \( 1 - 9.70e4iT - 6.97e9T^{2} \)
19 \( 1 - 2.09e5iT - 1.69e10T^{2} \)
23 \( 1 + 1.55e5T + 7.83e10T^{2} \)
29 \( 1 - 8.45e5T + 5.00e11T^{2} \)
31 \( 1 + 8.73e5iT - 8.52e11T^{2} \)
37 \( 1 + 1.13e6T + 3.51e12T^{2} \)
41 \( 1 - 1.80e6iT - 7.98e12T^{2} \)
43 \( 1 - 3.94e6T + 1.16e13T^{2} \)
47 \( 1 + 2.02e6iT - 2.38e13T^{2} \)
53 \( 1 + 1.13e7T + 6.22e13T^{2} \)
59 \( 1 + 9.30e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.87e7iT - 1.91e14T^{2} \)
67 \( 1 + 3.87e7T + 4.06e14T^{2} \)
71 \( 1 - 4.45e7T + 6.45e14T^{2} \)
73 \( 1 - 4.68e7iT - 8.06e14T^{2} \)
79 \( 1 + 3.24e6T + 1.51e15T^{2} \)
83 \( 1 + 7.02e7iT - 2.25e15T^{2} \)
89 \( 1 + 8.92e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.27e8iT - 7.83e15T^{2} \)
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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

−17.25825774414950123433568770979, −16.11512571772058438616721399170, −14.32633472413931279757462916286, −13.09771949204202446375127572995, −12.24855051228745930910965168794, −10.27880113191616934778718022472, −7.893278058301279733998532062666, −6.33203046282976399299985453299, −4.10577422569934297697525647882, −1.35052716356263521759968408162, 2.91607987279349019406377410548, 4.87248896752027032418138615843, 6.78542561783607115665751063069, 9.292000889738062764498072963176, 10.86269329183905322080813789172, 12.31862639609115919139896814344, 14.01432704599243781392048473263, 15.36238827191098747319991986867, 16.03071589366916182572453018844, 18.01610636431843148060770188688

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