Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.435 - 0.900i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.854 + 2.87i)3-s + 3i·4-s + (−4.57 − 2.01i)5-s + (1.42 + 2.63i)6-s + (−5.77 + 3.94i)7-s + (4.94 + 4.94i)8-s + (−7.53 − 4.91i)9-s + (−4.65 + 1.81i)10-s + 2.58i·11-s + (−8.62 − 2.56i)12-s + (8.94 + 8.94i)13-s + (−1.29 + 6.87i)14-s + (9.69 − 11.4i)15-s − 4.99·16-s + (0.581 + 0.581i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.353i)2-s + (−0.284 + 0.958i)3-s + 0.750i·4-s + (−0.915 − 0.402i)5-s + (0.238 + 0.439i)6-s + (−0.825 + 0.564i)7-s + (0.618 + 0.618i)8-s + (−0.837 − 0.546i)9-s + (−0.465 + 0.181i)10-s + 0.235i·11-s + (−0.718 − 0.213i)12-s + (0.687 + 0.687i)13-s + (−0.0924 + 0.491i)14-s + (0.646 − 0.763i)15-s − 0.312·16-s + (0.0342 + 0.0342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.435 - 0.900i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (62, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.435 - 0.900i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.537658 + 0.857050i\)
\(L(\frac12)\)  \(\approx\)  \(0.537658 + 0.857050i\)
\(L(2)\)   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 \( 1 + (0.854 - 2.87i)T \)
5 \( 1 + (4.57 + 2.01i)T \)
7 \( 1 + (5.77 - 3.94i)T \)
good2 \( 1 + (-0.707 + 0.707i)T - 4iT^{2} \)
11 \( 1 - 2.58iT - 121T^{2} \)
13 \( 1 + (-8.94 - 8.94i)T + 169iT^{2} \)
17 \( 1 + (-0.581 - 0.581i)T + 289iT^{2} \)
19 \( 1 - 16.5T + 361T^{2} \)
23 \( 1 + (-26.5 - 26.5i)T + 529iT^{2} \)
29 \( 1 + 11.5T + 841T^{2} \)
31 \( 1 + 30.8iT - 961T^{2} \)
37 \( 1 + (41.4 + 41.4i)T + 1.36e3iT^{2} \)
41 \( 1 + 17.1T + 1.68e3T^{2} \)
43 \( 1 + (25.1 - 25.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-45.2 - 45.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (-34.1 - 34.1i)T + 2.80e3iT^{2} \)
59 \( 1 + 47.6iT - 3.48e3T^{2} \)
61 \( 1 - 78.9iT - 3.72e3T^{2} \)
67 \( 1 + (-72.4 - 72.4i)T + 4.48e3iT^{2} \)
71 \( 1 + 49.0iT - 5.04e3T^{2} \)
73 \( 1 + (-26.0 - 26.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 75.8iT - 6.24e3T^{2} \)
83 \( 1 + (53.6 - 53.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 14.0iT - 7.92e3T^{2} \)
97 \( 1 + (25.9 - 25.9i)T - 9.40e3iT^{2} \)
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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

−13.65719600113482453404955081913, −12.58522358288910578437917910202, −11.72126453253888259989720832731, −11.09016717087511982605890558224, −9.452459602407238599129613192151, −8.674493517375120974848097251184, −7.22817697259111283402348790037, −5.44134118258553907755753535623, −4.10116427939317470885058391971, −3.22023649089247347666792604690, 0.72357733200806825756980922867, 3.35124504733449531519435395325, 5.21686618409417179869951748797, 6.58087011239751239142650295218, 7.14535410397616679544791953092, 8.529091656670237450245783684796, 10.31215025938571045663511953817, 11.05934597537128289841006317579, 12.31596076481823657240859206380, 13.33348179570851606855134764752

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