Properties

Label 2-14-7.5-c12-0-5
Degree $2$
Conductor $14$
Sign $0.998 + 0.0481i$
Analytic cond. $12.7959$
Root an. cond. $3.57713$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−22.6 + 39.1i)2-s + (1.10e3 − 638. i)3-s + (−1.02e3 − 1.77e3i)4-s + (2.48e4 + 1.43e4i)5-s + 5.78e4i·6-s + (−6.66e4 − 9.69e4i)7-s + 9.26e4·8-s + (5.50e5 − 9.53e5i)9-s + (−1.12e6 + 6.48e5i)10-s + (−3.18e5 − 5.51e5i)11-s + (−2.26e6 − 1.30e6i)12-s + 3.67e6i·13-s + (5.30e6 − 4.18e5i)14-s + 3.65e7·15-s + (−2.09e6 + 3.63e6i)16-s + (1.12e7 − 6.48e6i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (1.51 − 0.876i)3-s + (−0.249 − 0.433i)4-s + (1.58 + 0.916i)5-s + 1.23i·6-s + (−0.566 − 0.824i)7-s + 0.353·8-s + (1.03 − 1.79i)9-s + (−1.12 + 0.648i)10-s + (−0.179 − 0.311i)11-s + (−0.758 − 0.438i)12-s + 0.761i·13-s + (0.704 − 0.0555i)14-s + 3.21·15-s + (−0.125 + 0.216i)16-s + (0.465 − 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0481i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.998 + 0.0481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.998 + 0.0481i$
Analytic conductor: \(12.7959\)
Root analytic conductor: \(3.57713\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :6),\ 0.998 + 0.0481i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.91717 - 0.0702499i\)
\(L(\frac12)\) \(\approx\) \(2.91717 - 0.0702499i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (22.6 - 39.1i)T \)
7 \( 1 + (6.66e4 + 9.69e4i)T \)
good3 \( 1 + (-1.10e3 + 638. i)T + (2.65e5 - 4.60e5i)T^{2} \)
5 \( 1 + (-2.48e4 - 1.43e4i)T + (1.22e8 + 2.11e8i)T^{2} \)
11 \( 1 + (3.18e5 + 5.51e5i)T + (-1.56e12 + 2.71e12i)T^{2} \)
13 \( 1 - 3.67e6iT - 2.32e13T^{2} \)
17 \( 1 + (-1.12e7 + 6.48e6i)T + (2.91e14 - 5.04e14i)T^{2} \)
19 \( 1 + (-1.90e7 - 1.10e7i)T + (1.10e15 + 1.91e15i)T^{2} \)
23 \( 1 + (-5.48e7 + 9.49e7i)T + (-1.09e16 - 1.89e16i)T^{2} \)
29 \( 1 + 1.71e8T + 3.53e17T^{2} \)
31 \( 1 + (1.21e8 - 7.04e7i)T + (3.93e17 - 6.82e17i)T^{2} \)
37 \( 1 + (-3.10e8 + 5.37e8i)T + (-3.29e18 - 5.70e18i)T^{2} \)
41 \( 1 - 2.48e9iT - 2.25e19T^{2} \)
43 \( 1 + 1.21e10T + 3.99e19T^{2} \)
47 \( 1 + (5.32e9 + 3.07e9i)T + (5.80e19 + 1.00e20i)T^{2} \)
53 \( 1 + (-1.86e10 - 3.22e10i)T + (-2.45e20 + 4.25e20i)T^{2} \)
59 \( 1 + (5.05e10 - 2.91e10i)T + (8.89e20 - 1.54e21i)T^{2} \)
61 \( 1 + (3.42e10 + 1.97e10i)T + (1.32e21 + 2.29e21i)T^{2} \)
67 \( 1 + (-7.43e9 - 1.28e10i)T + (-4.09e21 + 7.08e21i)T^{2} \)
71 \( 1 - 3.43e10T + 1.64e22T^{2} \)
73 \( 1 + (2.13e11 - 1.23e11i)T + (1.14e22 - 1.98e22i)T^{2} \)
79 \( 1 + (-1.71e11 + 2.97e11i)T + (-2.95e22 - 5.11e22i)T^{2} \)
83 \( 1 + 3.79e11iT - 1.06e23T^{2} \)
89 \( 1 + (5.04e11 + 2.91e11i)T + (1.23e23 + 2.13e23i)T^{2} \)
97 \( 1 - 5.78e11iT - 6.93e23T^{2} \)
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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

−16.77980256475371868165452325358, −14.72251040553476751489565981232, −13.91580275960082069760416245577, −13.27195843738879567307587629620, −10.15974817927018434949542876364, −9.121271905412833556937666334273, −7.35928029840973982322590945392, −6.38865665891125101232054277725, −3.01020138595748226986749330174, −1.51830963586901920346201506164, 1.84063405768952910470948151267, 3.08203771262200572671053821272, 5.21653537323196481488857775097, 8.429826602358446468610632237091, 9.447928185844740071003824546380, 10.03746392469646524637555312579, 12.78906123794983376926760669441, 13.66664994345294608064183187925, 15.19028774711035518020927831395, 16.60234435991454049887852470341

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