Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.654 - 0.756i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.78 − 4.81i)3-s + (−1.52 + 2.64i)5-s + (−0.608 + 6.97i)7-s + (−10.9 + 18.9i)9-s + (−0.106 + 0.0612i)11-s + 4.11·13-s + 17.0·15-s + (17.8 − 10.3i)17-s + (−4.46 + 7.74i)19-s + (35.2 − 16.4i)21-s + (−7.51 + 13.0i)23-s + (7.82 + 13.5i)25-s + 71.8·27-s + 31.6i·29-s + (−23.0 + 13.2i)31-s + ⋯
L(s)  = 1  + (−0.926 − 1.60i)3-s + (−0.305 + 0.529i)5-s + (−0.0868 + 0.996i)7-s + (−1.21 + 2.10i)9-s + (−0.00963 + 0.00556i)11-s + 0.316·13-s + 1.13·15-s + (1.05 − 0.606i)17-s + (−0.235 + 0.407i)19-s + (1.67 − 0.783i)21-s + (−0.326 + 0.566i)23-s + (0.312 + 0.541i)25-s + 2.65·27-s + 1.09i·29-s + (−0.742 + 0.428i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.654 - 0.756i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (145, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.654 - 0.756i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.654555 + 0.299249i\)
\(L(\frac12)\)  \(\approx\)  \(0.654555 + 0.299249i\)
\(L(2)\)   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 \( 1 + (0.608 - 6.97i)T \)
good3 \( 1 + (2.78 + 4.81i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.52 - 2.64i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (0.106 - 0.0612i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 4.11T + 169T^{2} \)
17 \( 1 + (-17.8 + 10.3i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (4.46 - 7.74i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (7.51 - 13.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 31.6iT - 841T^{2} \)
31 \( 1 + (23.0 - 13.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (25.1 + 14.5i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 9.26iT - 1.68e3T^{2} \)
43 \( 1 - 45.3iT - 1.84e3T^{2} \)
47 \( 1 + (-68.6 - 39.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (55.0 - 31.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-14.2 - 24.6i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (12.6 - 21.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (65.4 - 37.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 2.81T + 5.04e3T^{2} \)
73 \( 1 + (-11.0 + 6.40i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-35.6 + 61.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 30.0T + 6.88e3T^{2} \)
89 \( 1 + (-15.3 - 8.83i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 26.1iT - 9.40e3T^{2} \)
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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

−12.23858695538294400765340122621, −11.44511303393631366336610901969, −10.63823551227880168238001077334, −9.033771547424477803642889695185, −7.81043900422583740437619139432, −7.11455202527574772574350211837, −6.03323961367717028289219528487, −5.32072764062885642130964283318, −3.00328865444154747236481550221, −1.50176122273856486509819863595, 0.46612297442672134060979550935, 3.64797322894021452582885952004, 4.35337634650535208631645821393, 5.39617413372742055895425109662, 6.50895210982796931040843074521, 8.090361496770534893869086020831, 9.226310140018891109235987646035, 10.22317414209934382414408158441, 10.69352474914638472228048862355, 11.70833579257618042638648072325

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