Properties

Label 2-14-7.5-c8-0-1
Degree $2$
Conductor $14$
Sign $0.866 - 0.498i$
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  + (5.65 − 9.79i)2-s + (−81.1 + 46.8i)3-s + (−63.9 − 110. i)4-s + (928. + 535. i)5-s + 1.06e3i·6-s + (2.21e3 + 931. i)7-s − 1.44e3·8-s + (1.11e3 − 1.92e3i)9-s + (1.05e4 − 6.06e3i)10-s + (312. + 541. i)11-s + (1.03e4 + 5.99e3i)12-s + 4.33e4i·13-s + (2.16e4 − 1.64e4i)14-s − 1.00e5·15-s + (−8.19e3 + 1.41e4i)16-s + (9.10e4 − 5.25e4i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−1.00 + 0.578i)3-s + (−0.249 − 0.433i)4-s + (1.48 + 0.857i)5-s + 0.818i·6-s + (0.921 + 0.388i)7-s − 0.353·8-s + (0.169 − 0.294i)9-s + (1.05 − 0.606i)10-s + (0.0213 + 0.0369i)11-s + (0.501 + 0.289i)12-s + 1.51i·13-s + (0.563 − 0.427i)14-s − 1.98·15-s + (−0.125 + 0.216i)16-s + (1.09 − 0.629i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.866 - 0.498i$
Analytic conductor: \(5.70330\)
Root analytic conductor: \(2.38815\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :4),\ 0.866 - 0.498i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.65463 + 0.441882i\)
\(L(\frac12)\) \(\approx\) \(1.65463 + 0.441882i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.65 + 9.79i)T \)
7 \( 1 + (-2.21e3 - 931. i)T \)
good3 \( 1 + (81.1 - 46.8i)T + (3.28e3 - 5.68e3i)T^{2} \)
5 \( 1 + (-928. - 535. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-312. - 541. i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 4.33e4iT - 8.15e8T^{2} \)
17 \( 1 + (-9.10e4 + 5.25e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (1.38e5 + 8.01e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (1.22e4 - 2.12e4i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 2.01e5T + 5.00e11T^{2} \)
31 \( 1 + (-7.57e5 + 4.37e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-8.30e5 + 1.43e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 4.64e5iT - 7.98e12T^{2} \)
43 \( 1 + 2.94e6T + 1.16e13T^{2} \)
47 \( 1 + (2.19e6 + 1.26e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (1.79e6 + 3.10e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-5.42e6 + 3.13e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (1.03e7 + 5.97e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (5.34e6 + 9.25e6i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 2.25e7T + 6.45e14T^{2} \)
73 \( 1 + (-4.20e7 + 2.43e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-2.70e7 + 4.69e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 2.28e7iT - 2.25e15T^{2} \)
89 \( 1 + (-7.46e7 - 4.30e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 1.34e8iT - 7.83e15T^{2} \)
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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

−17.81557464649403067181123006619, −16.77266418925755286829325182231, −14.74479503309559431488268775627, −13.75416801037365780639102680634, −11.72298931567224075154928013187, −10.75668464235490858723186448076, −9.527213724707014142327768658612, −6.23964846946892308889603243095, −4.91270221888839629060186733502, −2.08622226188505217827437433837, 1.18450388211463989832713757557, 5.19418556946566651612247671047, 6.11507787531486442945116852151, 8.204247710400935484621328944709, 10.34130895776376371127930086578, 12.33021810573450690334402437548, 13.26880383454072717850356254863, 14.69347566454627060504667973350, 16.84421808931269037751213111717, 17.30545163156010719148047306437

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