Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.426 + 0.904i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.408 + 0.408i)2-s + (−1.22 + 1.22i)3-s − 3.66i·4-s + (0.563 − 4.96i)5-s − 1.00·6-s + (−1.87 − 1.87i)7-s + (3.13 − 3.13i)8-s − 2.99i·9-s + (2.25 − 1.79i)10-s − 6.25·11-s + (4.49 + 4.49i)12-s + (16.4 − 16.4i)13-s − 1.52i·14-s + (5.39 + 6.77i)15-s − 12.1·16-s + (20.4 + 20.4i)17-s + ⋯
L(s)  = 1  + (0.204 + 0.204i)2-s + (−0.408 + 0.408i)3-s − 0.916i·4-s + (0.112 − 0.993i)5-s − 0.166·6-s + (−0.267 − 0.267i)7-s + (0.391 − 0.391i)8-s − 0.333i·9-s + (0.225 − 0.179i)10-s − 0.568·11-s + (0.374 + 0.374i)12-s + (1.26 − 1.26i)13-s − 0.109i·14-s + (0.359 + 0.451i)15-s − 0.756·16-s + (1.20 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.426 + 0.904i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.426 + 0.904i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.06289 - 0.673897i\)
\(L(\frac12)\)  \(\approx\)  \(1.06289 - 0.673897i\)
\(L(2)\)   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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (-0.563 + 4.96i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good2 \( 1 + (-0.408 - 0.408i)T + 4iT^{2} \)
11 \( 1 + 6.25T + 121T^{2} \)
13 \( 1 + (-16.4 + 16.4i)T - 169iT^{2} \)
17 \( 1 + (-20.4 - 20.4i)T + 289iT^{2} \)
19 \( 1 + 7.15iT - 361T^{2} \)
23 \( 1 + (12.0 - 12.0i)T - 529iT^{2} \)
29 \( 1 - 18.1iT - 841T^{2} \)
31 \( 1 + 33.3T + 961T^{2} \)
37 \( 1 + (-18.8 - 18.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 50.8T + 1.68e3T^{2} \)
43 \( 1 + (-53.3 + 53.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-46.9 - 46.9i)T + 2.20e3iT^{2} \)
53 \( 1 + (28.9 - 28.9i)T - 2.80e3iT^{2} \)
59 \( 1 + 10.0iT - 3.48e3T^{2} \)
61 \( 1 - 85.6T + 3.72e3T^{2} \)
67 \( 1 + (-11.9 - 11.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 20.8T + 5.04e3T^{2} \)
73 \( 1 + (-35.2 + 35.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 31.9iT - 6.24e3T^{2} \)
83 \( 1 + (-6.49 + 6.49i)T - 6.88e3iT^{2} \)
89 \( 1 + 145. iT - 7.92e3T^{2} \)
97 \( 1 + (-31.5 - 31.5i)T + 9.40e3iT^{2} \)
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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

−13.24197044566196884237653074699, −12.56082181916247792231834472005, −10.91940317877740641646874230307, −10.27585858537658420297940612572, −9.157950084479541835295340144395, −7.82798881161581992924280349455, −5.92573739658353469841569550381, −5.45440448369624912391955557197, −3.92748900564170234286877562463, −1.01484168047419156876720035858, 2.47851443057465957393736170049, 3.89593829537448730844907785001, 5.82281467009512456912294731220, 7.01755263297757537616122065046, 7.971593888939377350491531801972, 9.459741575547276801950269613366, 10.92632558789214410267874797582, 11.62044419595793377458186465165, 12.58827825185401125129026727328, 13.66458107921883372578884734751

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