Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.806 + 0.591i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.19 − 2.41i)3-s + (−0.0446 − 0.0257i)5-s + (6.12 + 3.39i)7-s + (7.20 − 12.4i)9-s + (0.894 + 1.55i)11-s + 5.87i·13-s − 0.249·15-s + (23.0 − 13.2i)17-s + (−22.8 − 13.1i)19-s + (33.8 − 0.603i)21-s + (−12.8 + 22.2i)23-s + (−12.4 − 21.6i)25-s − 26.1i·27-s − 27.1·29-s + (−25.7 + 14.8i)31-s + ⋯
L(s)  = 1  + (1.39 − 0.806i)3-s + (−0.00892 − 0.00515i)5-s + (0.874 + 0.484i)7-s + (0.800 − 1.38i)9-s + (0.0813 + 0.140i)11-s + 0.452i·13-s − 0.0166·15-s + (1.35 − 0.781i)17-s + (−1.20 − 0.693i)19-s + (1.61 − 0.0287i)21-s + (−0.558 + 0.966i)23-s + (−0.499 − 0.865i)25-s − 0.969i·27-s − 0.937·29-s + (−0.829 + 0.479i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.806 + 0.591i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (33, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.806 + 0.591i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.43443 - 0.796669i\)
\(L(\frac12)\)  \(\approx\)  \(2.43443 - 0.796669i\)
\(L(2)\)   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 \( 1 + (-6.12 - 3.39i)T \)
good3 \( 1 + (-4.19 + 2.41i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (0.0446 + 0.0257i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-0.894 - 1.55i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 5.87iT - 169T^{2} \)
17 \( 1 + (-23.0 + 13.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (22.8 + 13.1i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (12.8 - 22.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 27.1T + 841T^{2} \)
31 \( 1 + (25.7 - 14.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-30.8 + 53.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 65.7iT - 1.68e3T^{2} \)
43 \( 1 - 9.52T + 1.84e3T^{2} \)
47 \( 1 + (-61.2 - 35.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (4.86 + 8.42i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (54.3 - 31.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (66.1 + 38.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-51.5 - 89.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 90.1T + 5.04e3T^{2} \)
73 \( 1 + (28.8 - 16.6i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (32.4 - 56.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 29.1iT - 6.88e3T^{2} \)
89 \( 1 + (-18.7 - 10.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 123. iT - 9.40e3T^{2} \)
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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

−12.13087204676781953333260265354, −11.12551389611929535804044999973, −9.590977787815999849641846756327, −8.877358810267792243854505987349, −7.88202774665182144145007300926, −7.30937473908932804514233127212, −5.80571715598420086286266818870, −4.20945991198829534425954479501, −2.72252672323508552798108079157, −1.63971955801756715894419809430, 1.92630429750291607928702576293, 3.51608671109342517691176200331, 4.29417416324093986897810809818, 5.75141236545473068818207518678, 7.63008665250226595368159801196, 8.162362579301273350367807784241, 9.101704496708546025227845462393, 10.22517795471717672737836158162, 10.74572711265460877121214063548, 12.20103886134532328158896864261

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