Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.726 + 0.687i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.97 − 2.86i)3-s + (5.45 − 3.15i)5-s + (5.64 − 4.13i)7-s + (11.9 + 20.7i)9-s + (2.70 − 4.68i)11-s − 15.9i·13-s − 36.1·15-s + (−17.7 − 10.2i)17-s + (−11.7 + 6.79i)19-s + (−39.9 + 4.32i)21-s + (2.35 + 4.07i)23-s + (7.37 − 12.7i)25-s − 85.7i·27-s + 1.76·29-s + (11.9 + 6.87i)31-s + ⋯
L(s)  = 1  + (−1.65 − 0.956i)3-s + (1.09 − 0.630i)5-s + (0.807 − 0.590i)7-s + (1.33 + 2.30i)9-s + (0.245 − 0.426i)11-s − 1.22i·13-s − 2.41·15-s + (−1.04 − 0.602i)17-s + (−0.619 + 0.357i)19-s + (−1.90 + 0.206i)21-s + (0.102 + 0.177i)23-s + (0.294 − 0.510i)25-s − 3.17i·27-s + 0.0608·29-s + (0.383 + 0.221i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.726 + 0.687i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.726 + 0.687i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.374345 - 0.939600i\)
\(L(\frac12)\)  \(\approx\)  \(0.374345 - 0.939600i\)
\(L(2)\)   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 \( 1 + (-5.64 + 4.13i)T \)
good3 \( 1 + (4.97 + 2.86i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-5.45 + 3.15i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.70 + 4.68i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 15.9iT - 169T^{2} \)
17 \( 1 + (17.7 + 10.2i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (11.7 - 6.79i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-2.35 - 4.07i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 1.76T + 841T^{2} \)
31 \( 1 + (-11.9 - 6.87i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (5.23 + 9.06i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 11.2iT - 1.68e3T^{2} \)
43 \( 1 + 49.1T + 1.84e3T^{2} \)
47 \( 1 + (-9.02 + 5.20i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (16.0 - 27.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-57.2 - 33.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (27.6 - 15.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-49.2 + 85.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 61.7T + 5.04e3T^{2} \)
73 \( 1 + (-15.6 - 9.02i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-15.0 - 25.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 63.4iT - 6.88e3T^{2} \)
89 \( 1 + (-119. + 68.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 131. iT - 9.40e3T^{2} \)
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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

−11.64040977723014569763965086547, −10.84200799725992095191513607388, −10.10988142177099683765728248255, −8.528102930269887858428342323768, −7.39638116116358628539679400439, −6.33555719015994050565594294410, −5.48453897290925815049866507838, −4.71452665708206406374843582047, −1.85880789534000425918911589658, −0.69667925966916715460957959241, 1.93325317674117615646919904034, 4.26056025057735499813155612342, 5.08617169547158220571078190908, 6.25850075856689735304306710886, 6.69433849394940102417325499728, 8.862340711812098216647369296208, 9.747976719982404249483345197605, 10.58261600436101328639930938501, 11.32839718951799275685224756761, 11.97650758687940487284124777296

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