Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.476 + 0.879i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.71·2-s + 1.73i·3-s − 1.06·4-s − 2.23i·5-s − 2.96i·6-s + (−3.33 − 6.15i)7-s + 8.67·8-s − 2.99·9-s + 3.82i·10-s + 17.0·11-s − 1.85i·12-s − 16.3i·13-s + (5.70 + 10.5i)14-s + 3.87·15-s − 10.5·16-s − 13.4i·17-s + ⋯
L(s)  = 1  − 0.856·2-s + 0.577i·3-s − 0.267·4-s − 0.447i·5-s − 0.494i·6-s + (−0.476 − 0.879i)7-s + 1.08·8-s − 0.333·9-s + 0.382i·10-s + 1.54·11-s − 0.154i·12-s − 1.25i·13-s + (0.407 + 0.752i)14-s + 0.258·15-s − 0.661·16-s − 0.789i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.476 + 0.879i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (76, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.476 + 0.879i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.589371 - 0.351071i\)
\(L(\frac12)\)  \(\approx\)  \(0.589371 - 0.351071i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
5 \( 1 + 2.23iT \)
7 \( 1 + (3.33 + 6.15i)T \)
good2 \( 1 + 1.71T + 4T^{2} \)
11 \( 1 - 17.0T + 121T^{2} \)
13 \( 1 + 16.3iT - 169T^{2} \)
17 \( 1 + 13.4iT - 289T^{2} \)
19 \( 1 + 13.7iT - 361T^{2} \)
23 \( 1 + 16.6T + 529T^{2} \)
29 \( 1 - 32.1T + 841T^{2} \)
31 \( 1 - 6.74iT - 961T^{2} \)
37 \( 1 + 69.2T + 1.36e3T^{2} \)
41 \( 1 - 39.7iT - 1.68e3T^{2} \)
43 \( 1 - 43.2T + 1.84e3T^{2} \)
47 \( 1 + 40.1iT - 2.20e3T^{2} \)
53 \( 1 - 22.5T + 2.80e3T^{2} \)
59 \( 1 + 81.6iT - 3.48e3T^{2} \)
61 \( 1 + 14.9iT - 3.72e3T^{2} \)
67 \( 1 - 72.0T + 4.48e3T^{2} \)
71 \( 1 + 25.7T + 5.04e3T^{2} \)
73 \( 1 + 75.0iT - 5.32e3T^{2} \)
79 \( 1 - 80.0T + 6.24e3T^{2} \)
83 \( 1 - 102. iT - 6.88e3T^{2} \)
89 \( 1 - 128. iT - 7.92e3T^{2} \)
97 \( 1 - 159. iT - 9.40e3T^{2} \)
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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

−13.51691762361797376614421291020, −12.21498401294169748301838378050, −10.83947057254680013173016312671, −9.918182959075368769158422635295, −9.180752731009028830365177799120, −8.132205525996844642581483338199, −6.78081513946094830380073988072, −4.94596138871093350836627720209, −3.71819608584162075286781847470, −0.74302812743333406581924166557, 1.76217013701862250769654404008, 3.98858830840453437637118199938, 6.06950285543742041098284521514, 7.05154132280589498635999032010, 8.520664642534429188320299919221, 9.176974692258688377144487972410, 10.27775264784221036976690011368, 11.69956096711628459870745512656, 12.43514161224003638144151787695, 13.91154540819841126900185294093

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