Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.548 + 0.835i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.08 − 2.08i)2-s + (−1.22 − 1.22i)3-s − 4.73i·4-s + (0.137 − 4.99i)5-s − 5.11·6-s + (−1.87 + 1.87i)7-s + (−1.53 − 1.53i)8-s + 2.99i·9-s + (−10.1 − 10.7i)10-s + 2.70·11-s + (−5.79 + 5.79i)12-s + (−2.37 − 2.37i)13-s + 7.81i·14-s + (−6.28 + 5.95i)15-s + 12.5·16-s + (16.3 − 16.3i)17-s + ⋯
L(s)  = 1  + (1.04 − 1.04i)2-s + (−0.408 − 0.408i)3-s − 1.18i·4-s + (0.0274 − 0.999i)5-s − 0.853·6-s + (−0.267 + 0.267i)7-s + (−0.191 − 0.191i)8-s + 0.333i·9-s + (−1.01 − 1.07i)10-s + 0.245·11-s + (−0.483 + 0.483i)12-s + (−0.183 − 0.183i)13-s + 0.558i·14-s + (−0.419 + 0.396i)15-s + 0.782·16-s + (0.963 − 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.548 + 0.835i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.548 + 0.835i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.938328 - 1.73863i\)
\(L(\frac12)\)  \(\approx\)  \(0.938328 - 1.73863i\)
\(L(2)\)   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 \( 1 + (-0.137 + 4.99i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good2 \( 1 + (-2.08 + 2.08i)T - 4iT^{2} \)
11 \( 1 - 2.70T + 121T^{2} \)
13 \( 1 + (2.37 + 2.37i)T + 169iT^{2} \)
17 \( 1 + (-16.3 + 16.3i)T - 289iT^{2} \)
19 \( 1 - 9.18iT - 361T^{2} \)
23 \( 1 + (-21.4 - 21.4i)T + 529iT^{2} \)
29 \( 1 - 52.3iT - 841T^{2} \)
31 \( 1 + 5.01T + 961T^{2} \)
37 \( 1 + (-23.2 + 23.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 60.5T + 1.68e3T^{2} \)
43 \( 1 + (8.78 + 8.78i)T + 1.84e3iT^{2} \)
47 \( 1 + (-2.24 + 2.24i)T - 2.20e3iT^{2} \)
53 \( 1 + (25.6 + 25.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 100. iT - 3.48e3T^{2} \)
61 \( 1 + 82.1T + 3.72e3T^{2} \)
67 \( 1 + (65.1 - 65.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 22.8T + 5.04e3T^{2} \)
73 \( 1 + (5.38 + 5.38i)T + 5.32e3iT^{2} \)
79 \( 1 + 117. iT - 6.24e3T^{2} \)
83 \( 1 + (-85.5 - 85.5i)T + 6.88e3iT^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 + (55.1 - 55.1i)T - 9.40e3iT^{2} \)
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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

−12.93950220700715539638795116466, −12.19186957154946266582865222542, −11.58552806786985543980195339392, −10.28615874859247710650125336023, −9.042263067980356073453861654161, −7.48775515912754461033908817641, −5.65915910706761846790685923866, −4.88925842487275906464904257570, −3.27144889280572399955034583893, −1.39771994173501321168004238808, 3.37114797856695397493830978115, 4.60890492895327716035373668983, 6.03437542406545296364427160219, 6.71735268576006775525467755235, 7.904909292555111196894938362768, 9.786608561162512760052764873286, 10.73618528078430322758295132624, 11.98908704712905126479386652311, 13.15607312231106898636647088601, 14.12312878081741891884783972335

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