Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.707 + 0.707i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.554i·3-s − 4.57·5-s − 2.64i·7-s + 8.69·9-s − 15.7i·11-s + 8.57·13-s − 2.53i·15-s + 28.3·17-s − 6.33i·19-s + 1.46·21-s − 31.0i·23-s − 4.05·25-s + 9.81i·27-s − 0.846·29-s + 21.6i·31-s + ⋯
L(s)  = 1  + 0.184i·3-s − 0.915·5-s − 0.377i·7-s + 0.965·9-s − 1.43i·11-s + 0.659·13-s − 0.169i·15-s + 1.66·17-s − 0.333i·19-s + 0.0698·21-s − 1.35i·23-s − 0.162·25-s + 0.363i·27-s − 0.0291·29-s + 0.697i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.707 + 0.707i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.707 + 0.707i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.28660 - 0.532928i\)
\(L(\frac12)\)  \(\approx\)  \(1.28660 - 0.532928i\)
\(L(2)\)   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 \( 1 \)
7 \( 1 + 2.64iT \)
good3 \( 1 - 0.554iT - 9T^{2} \)
5 \( 1 + 4.57T + 25T^{2} \)
11 \( 1 + 15.7iT - 121T^{2} \)
13 \( 1 - 8.57T + 169T^{2} \)
17 \( 1 - 28.3T + 289T^{2} \)
19 \( 1 + 6.33iT - 361T^{2} \)
23 \( 1 + 31.0iT - 529T^{2} \)
29 \( 1 + 0.846T + 841T^{2} \)
31 \( 1 - 21.6iT - 961T^{2} \)
37 \( 1 + 33.6T + 1.36e3T^{2} \)
41 \( 1 - 66.9T + 1.68e3T^{2} \)
43 \( 1 + 44.8iT - 1.84e3T^{2} \)
47 \( 1 + 38.4iT - 2.20e3T^{2} \)
53 \( 1 + 14.8T + 2.80e3T^{2} \)
59 \( 1 + 5.80iT - 3.48e3T^{2} \)
61 \( 1 + 52.6T + 3.72e3T^{2} \)
67 \( 1 - 117. iT - 4.48e3T^{2} \)
71 \( 1 - 81.2iT - 5.04e3T^{2} \)
73 \( 1 + 47.8T + 5.32e3T^{2} \)
79 \( 1 + 57.4iT - 6.24e3T^{2} \)
83 \( 1 - 102. iT - 6.88e3T^{2} \)
89 \( 1 + 89.2T + 7.92e3T^{2} \)
97 \( 1 + 3.44T + 9.40e3T^{2} \)
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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

−11.88300074730888753813002409486, −10.87735111335992321386909772328, −10.18799622410495823920758640028, −8.803070589439741883471218967175, −7.952885458650635550347452249367, −6.95674587664764970930380862472, −5.64000370532332327092665889492, −4.18440325178984026885294169897, −3.32376393142946992140980011745, −0.885762572433443073667195869933, 1.55485861189045533252588261315, 3.51682571060593413834509512078, 4.59489274929040809600377695679, 5.99845179738711105366692464226, 7.52048340797495048801694595620, 7.73334466946236490577582336164, 9.363268678604661081338668404457, 10.09968730781178910315350606110, 11.36521551227446917082030615839, 12.27280707137665353767534448827

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