Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.146 + 0.989i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.31 − 1.91i)2-s + (−2.97 − 0.410i)3-s + (5.34 − 9.24i)4-s + (1.93 − 1.11i)5-s + (−10.6 + 4.32i)6-s + (−5.86 + 3.82i)7-s − 25.5i·8-s + (8.66 + 2.44i)9-s + (4.28 − 7.41i)10-s + (9.48 + 5.47i)11-s + (−19.6 + 25.2i)12-s − 3.75·13-s + (−12.1 + 23.9i)14-s + (−6.21 + 2.52i)15-s + (−27.6 − 47.9i)16-s + (12.6 + 7.30i)17-s + ⋯
L(s)  = 1  + (1.65 − 0.957i)2-s + (−0.990 − 0.136i)3-s + (1.33 − 2.31i)4-s + (0.387 − 0.223i)5-s + (−1.77 + 0.721i)6-s + (−0.837 + 0.546i)7-s − 3.19i·8-s + (0.962 + 0.271i)9-s + (0.428 − 0.741i)10-s + (0.862 + 0.497i)11-s + (−1.63 + 2.10i)12-s − 0.288·13-s + (−0.866 + 1.70i)14-s + (−0.414 + 0.168i)15-s + (−1.72 − 2.99i)16-s + (0.744 + 0.429i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.146 + 0.989i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.146 + 0.989i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.60892 - 1.86406i\)
\(L(\frac12)\)  \(\approx\)  \(1.60892 - 1.86406i\)
\(L(2)\)   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 \( 1 + (2.97 + 0.410i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (5.86 - 3.82i)T \)
good2 \( 1 + (-3.31 + 1.91i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (-9.48 - 5.47i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 3.75T + 169T^{2} \)
17 \( 1 + (-12.6 - 7.30i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.54 - 9.60i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (8.53 - 4.92i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 10.0iT - 841T^{2} \)
31 \( 1 + (12.0 - 20.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-19.1 - 33.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 67.5iT - 1.68e3T^{2} \)
43 \( 1 + 77.4T + 1.84e3T^{2} \)
47 \( 1 + (-4.91 + 2.83i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (59.9 + 34.6i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (45.4 + 26.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.77 - 11.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (10.6 - 18.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 25.6iT - 5.04e3T^{2} \)
73 \( 1 + (-12.4 + 21.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-55.3 - 95.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 102. iT - 6.88e3T^{2} \)
89 \( 1 + (62.5 - 36.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 6.23T + 9.40e3T^{2} \)
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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

−12.86183740505578524946593421514, −12.29704763679622729992779399407, −11.66298957145646617711321832880, −10.33513112979520056872778865658, −9.649079968571203109522603902073, −6.80476042053557953431746752241, −5.93122322551699085363949568665, −5.00511946124070689269498101461, −3.57965581007219360042785706593, −1.67662789714391363915844090031, 3.35502898680722201726809973274, 4.59925269172672382020128023149, 5.89223389842212138929214328294, 6.54631591480045566635913830449, 7.53920053296433802117123766872, 9.645327921880958375632861550125, 11.15145981609454306290666051815, 12.02966621060414206098429042589, 12.97877548927734542720564387997, 13.77486450375901102216175516091

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