Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.959 - 0.283i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.112·2-s + 1.73i·3-s − 3.98·4-s + 2.23i·5-s − 0.195i·6-s + (−6.71 − 1.98i)7-s + 0.902·8-s − 2.99·9-s − 0.252i·10-s − 15.8·11-s − 6.90i·12-s + 13.3i·13-s + (0.758 + 0.224i)14-s − 3.87·15-s + 15.8·16-s − 15.6i·17-s + ⋯
L(s)  = 1  − 0.0564·2-s + 0.577i·3-s − 0.996·4-s + 0.447i·5-s − 0.0326i·6-s + (−0.959 − 0.283i)7-s + 0.112·8-s − 0.333·9-s − 0.0252i·10-s − 1.44·11-s − 0.575i·12-s + 1.02i·13-s + (0.0541 + 0.0160i)14-s − 0.258·15-s + 0.990·16-s − 0.922i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.283i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.959 - 0.283i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (76, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.959 - 0.283i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0583761 + 0.403538i\)
\(L(\frac12)\)  \(\approx\)  \(0.0583761 + 0.403538i\)
\(L(2)\)   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 \( 1 - 1.73iT \)
5 \( 1 - 2.23iT \)
7 \( 1 + (6.71 + 1.98i)T \)
good2 \( 1 + 0.112T + 4T^{2} \)
11 \( 1 + 15.8T + 121T^{2} \)
13 \( 1 - 13.3iT - 169T^{2} \)
17 \( 1 + 15.6iT - 289T^{2} \)
19 \( 1 - 30.8iT - 361T^{2} \)
23 \( 1 - 3.63T + 529T^{2} \)
29 \( 1 - 14.5T + 841T^{2} \)
31 \( 1 - 11.3iT - 961T^{2} \)
37 \( 1 - 17.3T + 1.36e3T^{2} \)
41 \( 1 - 27.9iT - 1.68e3T^{2} \)
43 \( 1 + 12.1T + 1.84e3T^{2} \)
47 \( 1 + 80.5iT - 2.20e3T^{2} \)
53 \( 1 + 55.9T + 2.80e3T^{2} \)
59 \( 1 - 79.5iT - 3.48e3T^{2} \)
61 \( 1 - 94.5iT - 3.72e3T^{2} \)
67 \( 1 + 103.T + 4.48e3T^{2} \)
71 \( 1 + 113.T + 5.04e3T^{2} \)
73 \( 1 + 20.3iT - 5.32e3T^{2} \)
79 \( 1 + 1.27T + 6.24e3T^{2} \)
83 \( 1 - 19.7iT - 6.88e3T^{2} \)
89 \( 1 + 131. iT - 7.92e3T^{2} \)
97 \( 1 - 12.4iT - 9.40e3T^{2} \)
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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

−13.90971138260856678632651288319, −13.18714947771442294146899514338, −11.95396304852745266873048384748, −10.38270370717940827091478413134, −9.910688397786543126585667491203, −8.764900119862718863073434959475, −7.44300582144029800501847600718, −5.84912239384846311514974660937, −4.50033305824677525352445900492, −3.16369221078375360571947733378, 0.30646226579973574785856289536, 2.98845106546030290889479681782, 4.88751123307868311969223732529, 6.01095975706788352016015363332, 7.69271872363467094276096981207, 8.619879636700383674713616256321, 9.697362898276254421781582194490, 10.80126556816310096615567075991, 12.63080798498657497454641160542, 12.90530227988174125794544303223

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