Properties

Degree 2
Conductor $ 2 \cdot 7 $
Sign $0.605 + 0.795i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (2.5 − 4.33i)3-s + (−1.99 + 3.46i)4-s + (4.5 + 7.79i)5-s − 10·6-s + (−14 + 12.1i)7-s + 7.99·8-s + (0.999 + 1.73i)9-s + (9 − 15.5i)10-s + (28.5 − 49.3i)11-s + (10 + 17.3i)12-s − 70·13-s + (35 + 12.1i)14-s + 45.0·15-s + (−8 − 13.8i)16-s + (−25.5 + 44.1i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.481 − 0.833i)3-s + (−0.249 + 0.433i)4-s + (0.402 + 0.697i)5-s − 0.680·6-s + (−0.755 + 0.654i)7-s + 0.353·8-s + (0.0370 + 0.0641i)9-s + (0.284 − 0.492i)10-s + (0.781 − 1.35i)11-s + (0.240 + 0.416i)12-s − 1.49·13-s + (0.668 + 0.231i)14-s + 0.774·15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(14\)    =    \(2 \cdot 7\)
\( \varepsilon \)  =  $0.605 + 0.795i$
motivic weight  =  \(3\)
character  :  $\chi_{14} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 14,\ (\ :3/2),\ 0.605 + 0.795i)$
$L(2)$  $\approx$  $0.822479 - 0.407725i$
$L(\frac12)$  $\approx$  $0.822479 - 0.407725i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
7 \( 1 + (14 - 12.1i)T \)
good3 \( 1 + (-2.5 + 4.33i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-4.5 - 7.79i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-28.5 + 49.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 70T + 2.19e3T^{2} \)
17 \( 1 + (25.5 - 44.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (34.5 + 59.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 114T + 2.43e4T^{2} \)
31 \( 1 + (11.5 - 19.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-126.5 - 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 42T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (100.5 + 174. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-196.5 + 340. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (109.5 - 189. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-354.5 - 614. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (209.5 - 362. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 96T + 3.57e5T^{2} \)
73 \( 1 + (-156.5 + 271. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (230.5 + 399. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 588T + 5.71e5T^{2} \)
89 \( 1 + (-508.5 - 880. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.83e3T + 9.12e5T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.18029280225304998422878326827, −18.14594227112232935887102024855, −16.60088533217385288874447020367, −14.54147840467397320909262650282, −13.29749318027771018416356786078, −11.98950982979369141487438511313, −10.17866315908668252753946338987, −8.545662963510313431303726219308, −6.62525929767708568596055772994, −2.65328915382901394963039405638, 4.56837402176893666395382714770, 7.08174257950588709211683065132, 9.368781952174122019062954225025, 9.850467289346286709120806720003, 12.56656183963219221409419477896, 14.26816558156304400322235008345, 15.43417976203765044553228414374, 16.69027607437785497004687294467, 17.62199509974744388632710877392, 19.70060149722936634821691663140

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