Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.967 + 0.251i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.91 + 0.511i)2-s + (−1.67 − 0.448i)3-s + (−0.0771 + 0.0445i)4-s + (−4.99 − 0.258i)5-s + 3.42·6-s + (6.99 + 0.240i)7-s + (5.71 − 5.71i)8-s + (2.59 + 1.50i)9-s + (9.67 − 2.06i)10-s + (1.58 + 2.74i)11-s + (0.148 − 0.0399i)12-s + (11.0 − 11.0i)13-s + (−13.4 + 3.12i)14-s + (8.23 + 2.67i)15-s + (−7.81 + 13.5i)16-s + (4.27 − 15.9i)17-s + ⋯
L(s)  = 1  + (−0.955 + 0.255i)2-s + (−0.557 − 0.149i)3-s + (−0.0192 + 0.0111i)4-s + (−0.998 − 0.0516i)5-s + 0.570·6-s + (0.999 + 0.0343i)7-s + (0.714 − 0.714i)8-s + (0.288 + 0.166i)9-s + (0.967 − 0.206i)10-s + (0.144 + 0.249i)11-s + (0.0124 − 0.00332i)12-s + (0.850 − 0.850i)13-s + (−0.963 + 0.222i)14-s + (0.549 + 0.178i)15-s + (−0.488 + 0.846i)16-s + (0.251 − 0.938i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.967 + 0.251i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.967 + 0.251i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.598360 - 0.0765371i\)
\(L(\frac12)\)  \(\approx\)  \(0.598360 - 0.0765371i\)
\(L(2)\)   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 \( 1 + (1.67 + 0.448i)T \)
5 \( 1 + (4.99 + 0.258i)T \)
7 \( 1 + (-6.99 - 0.240i)T \)
good2 \( 1 + (1.91 - 0.511i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (-1.58 - 2.74i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-11.0 + 11.0i)T - 169iT^{2} \)
17 \( 1 + (-4.27 + 15.9i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-24.9 - 14.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (2.86 + 10.6i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 20.9iT - 841T^{2} \)
31 \( 1 + (30.4 + 52.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (4.96 - 1.32i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 0.605T + 1.68e3T^{2} \)
43 \( 1 + (-16.0 + 16.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (-34.1 + 9.14i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-60.8 - 16.2i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-88.6 + 51.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (16.9 - 29.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (13.4 - 50.2i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 25.3T + 5.04e3T^{2} \)
73 \( 1 + (86.1 + 23.0i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-6.66 - 3.84i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-81.1 + 81.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (35.6 + 20.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-1.74 - 1.74i)T + 9.40e3iT^{2} \)
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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

−13.38768057842872499441289390785, −12.17240951562034646315866087902, −11.30285531251863292419105838534, −10.30395281396854237853151493169, −8.931095645719484493751849973829, −7.86865209863193006638726989752, −7.29836575194214987572603341791, −5.37025772323663401867529340671, −3.94606502643475957377560698956, −0.883088260058526439860057423959, 1.22132687042244611180142399003, 4.03689329095051831787445493645, 5.32208963275622852982254308051, 7.19179375973985874536531880500, 8.272275259118492804886428131745, 9.142303784320880469224119978044, 10.60088735612720213836580379644, 11.25797172733977314017374081772, 11.97154213859831927800287680464, 13.64907682315353595883154078061

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