Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.516 + 0.856i$
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 + (2.87 + 0.854i)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 + (2.56 − 8.62i)12-s + (8.94 − 8.94i)13-s + (1.29 + 6.87i)14-s + (14.8 − 1.86i)15-s − 4.99·16-s + (−0.581 + 0.581i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.353i)2-s + (0.958 + 0.284i)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 + (0.213 − 0.718i)12-s + (0.687 − 0.687i)13-s + (0.0924 + 0.491i)14-s + (0.992 − 0.124i)15-s − 0.312·16-s + (−0.0342 + 0.0342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.516 + 0.856i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.516 + 0.856i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.38240 - 0.780554i\)
\(L(\frac12)\)  \(\approx\)  \(1.38240 - 0.780554i\)
\(L(2)\)   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 \( 1 + (-2.87 - 0.854i)T \)
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

−13.75589292851594741574753455570, −12.50175008620243634751576170572, −10.74640791052954079907739313475, −9.878228763263182295776438532291, −9.430060226345861584943311614799, −8.175879324064231660947216955195, −6.50071961200445837347581765508, −5.14567620203357335433696521188, −3.25435627113390424588350071741, −1.52106135502945863759403470020, 2.43491242052798933074338252522, 3.66503462519461411284981128671, 6.15962416679526582048365170053, 7.00316951717655618268308274933, 8.372978752696680162987995488712, 9.184268348007108671024993485674, 10.02465662300737648798674802162, 11.84290969418692371221184213413, 12.91577817761280368293462481134, 13.63680828105378219226818870002

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