Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.423 + 0.906i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.41 − 2.41i)2-s + (1.22 + 1.22i)3-s − 7.68i·4-s + (4.18 + 2.74i)5-s + 5.92·6-s + (−1.87 + 1.87i)7-s + (−8.90 − 8.90i)8-s + 2.99i·9-s + (16.7 − 3.47i)10-s − 20.9·11-s + (9.40 − 9.40i)12-s + (−9.34 − 9.34i)13-s + 9.04i·14-s + (1.76 + 8.47i)15-s − 12.2·16-s + (7.08 − 7.08i)17-s + ⋯
L(s)  = 1  + (1.20 − 1.20i)2-s + (0.408 + 0.408i)3-s − 1.92i·4-s + (0.836 + 0.548i)5-s + 0.986·6-s + (−0.267 + 0.267i)7-s + (−1.11 − 1.11i)8-s + 0.333i·9-s + (1.67 − 0.347i)10-s − 1.90·11-s + (0.784 − 0.784i)12-s + (−0.718 − 0.718i)13-s + 0.645i·14-s + (0.117 + 0.565i)15-s − 0.768·16-s + (0.416 − 0.416i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.423 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.423 + 0.906i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.423 + 0.906i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.27524 - 1.44869i\)
\(L(\frac12)\)  \(\approx\)  \(2.27524 - 1.44869i\)
\(L(2)\)   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 \( 1 + (-4.18 - 2.74i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good2 \( 1 + (-2.41 + 2.41i)T - 4iT^{2} \)
11 \( 1 + 20.9T + 121T^{2} \)
13 \( 1 + (9.34 + 9.34i)T + 169iT^{2} \)
17 \( 1 + (-7.08 + 7.08i)T - 289iT^{2} \)
19 \( 1 - 14.9iT - 361T^{2} \)
23 \( 1 + (-12.9 - 12.9i)T + 529iT^{2} \)
29 \( 1 + 39.6iT - 841T^{2} \)
31 \( 1 - 12.8T + 961T^{2} \)
37 \( 1 + (31.7 - 31.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 69.4T + 1.68e3T^{2} \)
43 \( 1 + (-4.46 - 4.46i)T + 1.84e3iT^{2} \)
47 \( 1 + (4.41 - 4.41i)T - 2.20e3iT^{2} \)
53 \( 1 + (48.5 + 48.5i)T + 2.80e3iT^{2} \)
59 \( 1 + 29.4iT - 3.48e3T^{2} \)
61 \( 1 - 7.09T + 3.72e3T^{2} \)
67 \( 1 + (1.39 - 1.39i)T - 4.48e3iT^{2} \)
71 \( 1 + 15.9T + 5.04e3T^{2} \)
73 \( 1 + (-32.4 - 32.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 66.1iT - 6.24e3T^{2} \)
83 \( 1 + (83.6 + 83.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 62.7iT - 7.92e3T^{2} \)
97 \( 1 + (85.4 - 85.4i)T - 9.40e3iT^{2} \)
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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

−13.25212365839498520003372010256, −12.57991444428888016348215566664, −11.20614153847878511595597189363, −10.18464599616167797957367840178, −9.815188435231095728308674272826, −7.79630764081569523186737594729, −5.77935658930368348780263784906, −4.99178253132589233973698800634, −3.15326896612943940512535457753, −2.40725196670167229761268255753, 2.74107191830797683522933022906, 4.66551532386580781972595069423, 5.59047950560194448326105865335, 6.85624248621104130967961645817, 7.78582583889129788743266355622, 9.024458795645925743736367181912, 10.49399737913839283705099958377, 12.55840631687303414624515075296, 12.86982204409400124173251614537, 13.80219820560027324775434381903

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