Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.999 - 0.0168i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.70 − 2.94i)3-s + (2.15 + 3.73i)5-s + (1.43 + 6.85i)7-s + (−1.28 − 2.23i)9-s + (15.4 + 8.94i)11-s + 3.25·13-s + 14.6·15-s + (−13.6 − 7.86i)17-s + (−0.778 − 1.34i)19-s + (22.6 + 7.43i)21-s + (−20.7 − 35.8i)23-s + (3.18 − 5.50i)25-s + 21.8·27-s − 3.74i·29-s + (0.0145 + 0.00838i)31-s + ⋯
L(s)  = 1  + (0.567 − 0.982i)3-s + (0.431 + 0.747i)5-s + (0.204 + 0.978i)7-s + (−0.143 − 0.248i)9-s + (1.40 + 0.813i)11-s + 0.250·13-s + 0.979·15-s + (−0.801 − 0.462i)17-s + (−0.0409 − 0.0709i)19-s + (1.07 + 0.354i)21-s + (−0.900 − 1.55i)23-s + (0.127 − 0.220i)25-s + 0.809·27-s − 0.129i·29-s + (0.000468 + 0.000270i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0168i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.999 - 0.0168i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.999 - 0.0168i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.11830 + 0.0177990i\)
\(L(\frac12)\)  \(\approx\)  \(2.11830 + 0.0177990i\)
\(L(2)\)   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 \( 1 + (-1.43 - 6.85i)T \)
good3 \( 1 + (-1.70 + 2.94i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-2.15 - 3.73i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-15.4 - 8.94i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 3.25T + 169T^{2} \)
17 \( 1 + (13.6 + 7.86i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (0.778 + 1.34i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (20.7 + 35.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 3.74iT - 841T^{2} \)
31 \( 1 + (-0.0145 - 0.00838i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (1.16 - 0.674i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 70.3iT - 1.68e3T^{2} \)
43 \( 1 + 13.0iT - 1.84e3T^{2} \)
47 \( 1 + (30.9 - 17.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-39.7 - 22.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-34.3 + 59.4i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (48.0 + 83.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (12.0 + 6.97i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 75.7T + 5.04e3T^{2} \)
73 \( 1 + (46.0 + 26.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (11.6 + 20.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 102.T + 6.88e3T^{2} \)
89 \( 1 + (76.6 - 44.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 140. iT - 9.40e3T^{2} \)
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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

−12.18067083623868668334660067335, −11.24976548934090565820868745422, −9.960747736207166867747188680346, −8.954092050041647999186859151163, −8.075513410721802435388958194297, −6.75145250054567586042997618791, −6.34894549459050887585974915374, −4.54007158457306252829906101620, −2.69483709667931724450900695614, −1.80634318410256910302443363107, 1.35557436274588786896393274427, 3.64682169333829377915720011182, 4.22309359471935586998795689924, 5.65264269464528784419171375197, 6.96152765876726820718798561440, 8.462336961158835991483889926044, 9.104657260478273864967838706519, 9.920548821412674957090005253668, 10.91317913859502215171082949211, 11.89770015110680036194107931039

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