Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.935 + 0.353i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.41·3-s + 1.54i·5-s − 2.64i·7-s + 2.65·9-s + 4.48·11-s − 1.54i·13-s − 5.29i·15-s + 23.6·17-s + 24.8·19-s + 9.03i·21-s − 35.2i·23-s + 22.5·25-s + 21.6·27-s − 22.4i·29-s + 46.7i·31-s + ⋯
L(s)  = 1  − 1.13·3-s + 0.309i·5-s − 0.377i·7-s + 0.295·9-s + 0.407·11-s − 0.119i·13-s − 0.352i·15-s + 1.39·17-s + 1.30·19-s + 0.430i·21-s − 1.53i·23-s + 0.903·25-s + 0.802·27-s − 0.774i·29-s + 1.50i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.935 + 0.353i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (15, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.935 + 0.353i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.03940 - 0.189873i\)
\(L(\frac12)\)  \(\approx\)  \(1.03940 - 0.189873i\)
\(L(2)\)   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 \( 1 + 3.41T + 9T^{2} \)
5 \( 1 - 1.54iT - 25T^{2} \)
11 \( 1 - 4.48T + 121T^{2} \)
13 \( 1 + 1.54iT - 169T^{2} \)
17 \( 1 - 23.6T + 289T^{2} \)
19 \( 1 - 24.8T + 361T^{2} \)
23 \( 1 + 35.2iT - 529T^{2} \)
29 \( 1 + 22.4iT - 841T^{2} \)
31 \( 1 - 46.7iT - 961T^{2} \)
37 \( 1 + 58.5iT - 1.36e3T^{2} \)
41 \( 1 + 26.9T + 1.68e3T^{2} \)
43 \( 1 - 17.1T + 1.84e3T^{2} \)
47 \( 1 - 36.1iT - 2.20e3T^{2} \)
53 \( 1 - 97.8iT - 2.80e3T^{2} \)
59 \( 1 + 61.5T + 3.48e3T^{2} \)
61 \( 1 + 37.6iT - 3.72e3T^{2} \)
67 \( 1 - 33.3T + 4.48e3T^{2} \)
71 \( 1 + 102. iT - 5.04e3T^{2} \)
73 \( 1 - 69.3T + 5.32e3T^{2} \)
79 \( 1 + 38.7iT - 6.24e3T^{2} \)
83 \( 1 + 3.61T + 6.88e3T^{2} \)
89 \( 1 - 44.0T + 7.92e3T^{2} \)
97 \( 1 - 96.1T + 9.40e3T^{2} \)
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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

−12.06389373932907009263302238933, −10.90804068601203006943185160966, −10.38984120578824720990870369782, −9.174265539905812216928716626495, −7.77614335464481283665981319140, −6.73846378729020161694477070503, −5.77676430335292033363517752446, −4.74421740273401381882412873870, −3.17456994781437947257308890850, −0.883292174761147313084179093256, 1.13927552815464686216722861428, 3.36152409000113643669722445606, 5.08189643313897293910187252361, 5.65678107773337871540974453576, 6.85532526023514157462410474096, 8.045156681462218737076019202083, 9.330792155172143027008770880768, 10.16724332921635795686709323611, 11.55948184303604530988278149467, 11.73563472654870568805578499305

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