Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.810 + 0.585i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.88 − 1.88i)2-s + (−1.39 − 2.65i)3-s − 3.12i·4-s + (−4.96 − 0.551i)5-s + (−7.64 − 2.36i)6-s + (2.09 − 6.68i)7-s + (1.65 + 1.65i)8-s + (−5.08 + 7.42i)9-s + (−10.4 + 8.33i)10-s − 17.9i·11-s + (−8.28 + 4.36i)12-s + (11.1 + 11.1i)13-s + (−8.66 − 16.5i)14-s + (5.48 + 13.9i)15-s + 18.7·16-s + (0.666 + 0.666i)17-s + ⋯
L(s)  = 1  + (0.943 − 0.943i)2-s + (−0.466 − 0.884i)3-s − 0.780i·4-s + (−0.993 − 0.110i)5-s + (−1.27 − 0.394i)6-s + (0.298 − 0.954i)7-s + (0.207 + 0.207i)8-s + (−0.564 + 0.825i)9-s + (−1.04 + 0.833i)10-s − 1.62i·11-s + (−0.690 + 0.363i)12-s + (0.856 + 0.856i)13-s + (−0.618 − 1.18i)14-s + (0.365 + 0.930i)15-s + 1.17·16-s + (0.0392 + 0.0392i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.810 + 0.585i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (62, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.810 + 0.585i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.504504 - 1.56153i\)
\(L(\frac12)\)  \(\approx\)  \(0.504504 - 1.56153i\)
\(L(2)\)   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 \( 1 + (1.39 + 2.65i)T \)
5 \( 1 + (4.96 + 0.551i)T \)
7 \( 1 + (-2.09 + 6.68i)T \)
good2 \( 1 + (-1.88 + 1.88i)T - 4iT^{2} \)
11 \( 1 + 17.9iT - 121T^{2} \)
13 \( 1 + (-11.1 - 11.1i)T + 169iT^{2} \)
17 \( 1 + (-0.666 - 0.666i)T + 289iT^{2} \)
19 \( 1 - 10.8T + 361T^{2} \)
23 \( 1 + (2.20 + 2.20i)T + 529iT^{2} \)
29 \( 1 + 22.9T + 841T^{2} \)
31 \( 1 + 26.1iT - 961T^{2} \)
37 \( 1 + (-41.6 - 41.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 6.85T + 1.68e3T^{2} \)
43 \( 1 + (37.6 - 37.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (5.55 + 5.55i)T + 2.20e3iT^{2} \)
53 \( 1 + (-32.4 - 32.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 99.8iT - 3.48e3T^{2} \)
61 \( 1 - 44.6iT - 3.72e3T^{2} \)
67 \( 1 + (18.0 + 18.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 6.35iT - 5.04e3T^{2} \)
73 \( 1 + (55.3 + 55.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 59.7iT - 6.24e3T^{2} \)
83 \( 1 + (-42.2 + 42.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 58.4iT - 7.92e3T^{2} \)
97 \( 1 + (11.1 - 11.1i)T - 9.40e3iT^{2} \)
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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

−13.34028182027925549836995049929, −11.77496779115403251856401190399, −11.45018988637742528985734351252, −10.70752832138126419016653556893, −8.437554856706355187901712551680, −7.51320284450398602225385630577, −6.00659010180870464064359026403, −4.46841989942011735141908209718, −3.30160068283432219414985128217, −1.11545248726907422777713751183, 3.60177215926376290440302520181, 4.77102020032507692533914816596, 5.62894076947209545276705711901, 6.99320016066445127578493745420, 8.219028483365831272757160897616, 9.683224319381976208721191080981, 10.94386717892081727366377835650, 12.06496833677264620190103672144, 12.82130165896110152629372499330, 14.49504134338378688566585329311

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