Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.796 - 0.604i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.796 - 0.604i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (145, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.796 - 0.604i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.00358522 + 0.0106624i\)
\(L(\frac12)\)  \(\approx\)  \(0.00358522 + 0.0106624i\)
\(L(2)\)   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

−11.39549848783675663787453714337, −10.70826217392226496877516653736, −9.746721054683644065438602837420, −7.897630956017236910274893099194, −7.37655278859016576508321905332, −6.59922908246679090148255141632, −5.21163995023776014767460958493, −3.73717637782344577223649334797, −1.95853014266626727549651834080, −0.00619307178390111354362978376, 2.65936518752057519382925940066, 4.48128494603854242837625507383, 5.09044575291215511703973584688, 6.13733441377627187618935721672, 7.985141385570668457953865242946, 8.645546819515640638803541655369, 9.793369793178796552544969728046, 10.74799221886202023893139667785, 11.51919993221209875445744044595, 12.47504230795074487752024833408

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