Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.988 - 0.148i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.949 + 1.64i)2-s + (2.65 − 1.40i)3-s + (0.196 − 0.340i)4-s + (0.399 − 4.98i)5-s + (4.82 + 3.02i)6-s + (−4.60 + 5.26i)7-s + 8.34·8-s + (5.04 − 7.45i)9-s + (8.57 − 4.07i)10-s + (−8.35 − 4.82i)11-s + (0.0423 − 1.17i)12-s + 16.9i·13-s + (−13.0 − 2.57i)14-s + (−5.94 − 13.7i)15-s + (7.13 + 12.3i)16-s + (−12.1 + 21.1i)17-s + ⋯
L(s)  = 1  + (0.474 + 0.822i)2-s + (0.883 − 0.468i)3-s + (0.0490 − 0.0850i)4-s + (0.0799 − 0.996i)5-s + (0.804 + 0.504i)6-s + (−0.658 + 0.752i)7-s + 1.04·8-s + (0.560 − 0.827i)9-s + (0.857 − 0.407i)10-s + (−0.759 − 0.438i)11-s + (0.00353 − 0.0981i)12-s + 1.30i·13-s + (−0.931 − 0.184i)14-s + (−0.396 − 0.918i)15-s + (0.446 + 0.772i)16-s + (−0.716 + 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.988 - 0.148i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (74, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.988 - 0.148i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.19468 + 0.163754i\)
\(L(\frac12)\)  \(\approx\)  \(2.19468 + 0.163754i\)
\(L(2)\)   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 \( 1 + (-2.65 + 1.40i)T \)
5 \( 1 + (-0.399 + 4.98i)T \)
7 \( 1 + (4.60 - 5.26i)T \)
good2 \( 1 + (-0.949 - 1.64i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (8.35 + 4.82i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 16.9iT - 169T^{2} \)
17 \( 1 + (12.1 - 21.1i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.95 - 12.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (0.354 + 0.614i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 16.5iT - 841T^{2} \)
31 \( 1 + (-7.12 + 12.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (1.08 - 0.626i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 24.1iT - 1.68e3T^{2} \)
43 \( 1 + 57.6iT - 1.84e3T^{2} \)
47 \( 1 + (16.0 + 27.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-8.67 + 15.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (75.8 + 43.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-52.2 - 90.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-86.1 - 49.7i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 50.7iT - 5.04e3T^{2} \)
73 \( 1 + (81.5 + 47.0i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (3.71 + 6.43i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 69.6T + 6.88e3T^{2} \)
89 \( 1 + (-78.3 + 45.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 90.4iT - 9.40e3T^{2} \)
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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

−13.50421473769890785499942212903, −13.00350670535452213555197251783, −11.83775331206096983829080858055, −10.03787770802063919260472344353, −8.893574084359446267075408203816, −8.045864308849611478340429677565, −6.64362508531797206542521307318, −5.67648714672709117777227426704, −4.11946729931104834648632794084, −1.95696981453703377473036784161, 2.66125117462803068547134555250, 3.30834957273840650663355511119, 4.78278408134890098397866452442, 7.06026938854340389526027809473, 7.79941973066969503609195794003, 9.631079360291896521288544024019, 10.46066037249972862241507410491, 11.11747791796493526856962279146, 12.74195418452380776173270769230, 13.44816549454241828854165849297

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