Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.330 - 0.943i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 − 3.44i)3-s + (−1.63 − 0.941i)5-s + (−5.14 + 4.74i)7-s + (−3.42 + 5.93i)9-s + (−3.93 − 6.82i)11-s + 11.4i·13-s + 7.49i·15-s + (1.44 + 2.51i)17-s + (−15.0 + 26.0i)19-s + (26.6 + 8.30i)21-s + (33.3 + 19.2i)23-s + (−10.7 − 18.5i)25-s − 8.56·27-s + 27.8i·29-s + (−19.4 + 11.2i)31-s + ⋯
L(s)  = 1  + (−0.663 − 1.14i)3-s + (−0.326 − 0.188i)5-s + (−0.735 + 0.677i)7-s + (−0.380 + 0.659i)9-s + (−0.358 − 0.620i)11-s + 0.883i·13-s + 0.499i·15-s + (0.0852 + 0.147i)17-s + (−0.790 + 1.36i)19-s + (1.26 + 0.395i)21-s + (1.45 + 0.838i)23-s + (−0.429 − 0.743i)25-s − 0.317·27-s + 0.961i·29-s + (−0.628 + 0.362i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.330 - 0.943i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (79, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.330 - 0.943i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.114099 + 0.160843i\)
\(L(\frac12)\)  \(\approx\)  \(0.114099 + 0.160843i\)
\(L(2)\)   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 \( 1 + (5.14 - 4.74i)T \)
good3 \( 1 + (1.99 + 3.44i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.63 + 0.941i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (3.93 + 6.82i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 11.4iT - 169T^{2} \)
17 \( 1 + (-1.44 - 2.51i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (15.0 - 26.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-33.3 - 19.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 27.8iT - 841T^{2} \)
31 \( 1 + (19.4 - 11.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (39.4 + 22.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 40.6T + 1.68e3T^{2} \)
43 \( 1 - 47.2T + 1.84e3T^{2} \)
47 \( 1 + (71.5 + 41.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (23.2 - 13.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (5.20 + 9.01i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (19.1 + 11.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (29.6 + 51.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 38.2iT - 5.04e3T^{2} \)
73 \( 1 + (6.98 + 12.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (44.3 + 25.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 89.4T + 6.88e3T^{2} \)
89 \( 1 + (52.6 - 91.1i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 55.3T + 9.40e3T^{2} \)
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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

−12.37850684391577976835498539325, −11.64297139042004761848547345539, −10.61836547056759986979243414451, −9.241947472782703152476926287902, −8.280151012149565549724775771620, −7.09247419343209938405354492983, −6.27636621369410391739695933699, −5.34078261083037930674506359481, −3.50665329629585684070345654213, −1.73015094523711141262642636064, 0.11371656378329650363855742507, 3.06400413257715554281916241307, 4.32153419772836824612652165276, 5.19348863100775487264698279919, 6.57519167934701808881744041836, 7.59372295734976459190687509535, 9.093475797654389263243199354224, 10.02714378500329993109797904842, 10.69705755951090605903108395346, 11.37992147757783827915177516854

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