Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.940 + 0.339i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.61 + 1.51i)2-s + (−0.469 − 2.96i)3-s + (2.56 − 4.43i)4-s + (1.93 − 1.11i)5-s + (5.70 + 7.04i)6-s + (−4.99 + 4.90i)7-s + 3.39i·8-s + (−8.55 + 2.78i)9-s + (−3.37 + 5.84i)10-s + (1.06 + 0.614i)11-s + (−14.3 − 5.50i)12-s − 18.8·13-s + (5.65 − 20.3i)14-s + (−4.22 − 5.21i)15-s + (5.11 + 8.86i)16-s + (−16.5 − 9.57i)17-s + ⋯
L(s)  = 1  + (−1.30 + 0.755i)2-s + (−0.156 − 0.987i)3-s + (0.640 − 1.10i)4-s + (0.387 − 0.223i)5-s + (0.950 + 1.17i)6-s + (−0.713 + 0.700i)7-s + 0.424i·8-s + (−0.951 + 0.309i)9-s + (−0.337 + 0.584i)10-s + (0.0967 + 0.0558i)11-s + (−1.19 − 0.459i)12-s − 1.44·13-s + (0.404 − 1.45i)14-s + (−0.281 − 0.347i)15-s + (0.319 + 0.554i)16-s + (−0.975 − 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.940 + 0.339i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.940 + 0.339i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0136482 - 0.0779857i\)
\(L(\frac12)\)  \(\approx\)  \(0.0136482 - 0.0779857i\)
\(L(2)\)   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 \( 1 + (0.469 + 2.96i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (4.99 - 4.90i)T \)
good2 \( 1 + (2.61 - 1.51i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (-1.06 - 0.614i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 18.8T + 169T^{2} \)
17 \( 1 + (16.5 + 9.57i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (10.0 + 17.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (16.4 - 9.46i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 31.9iT - 841T^{2} \)
31 \( 1 + (14.6 - 25.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (9.97 + 17.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 58.4iT - 1.68e3T^{2} \)
43 \( 1 - 57.9T + 1.84e3T^{2} \)
47 \( 1 + (-41.1 + 23.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (3.20 + 1.85i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-22.3 - 12.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (41.8 + 72.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (21.3 - 37.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 0.779iT - 5.04e3T^{2} \)
73 \( 1 + (-4.35 + 7.54i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (31.2 + 54.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 75.5iT - 6.88e3T^{2} \)
89 \( 1 + (73.5 - 42.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 133.T + 9.40e3T^{2} \)
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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

−12.90478754280858367332184146943, −12.16605199607317367296110276950, −10.65903184487639449117790463697, −9.308196016601315848632585016836, −8.794021233615678035962450195077, −7.34217046070586030600104182351, −6.69967012030467023031701282166, −5.44920668931065265017399063128, −2.27538704337039420098665382761, −0.083608542765862247376480021915, 2.52842015144385513194817053636, 4.22127762740060344972337048425, 6.12940279112938932733015887199, 7.74944692457873820285161300328, 9.074026710074435023786443887105, 9.951280119781593398677912571630, 10.36846095827709797630158659489, 11.41358759696169859820655140876, 12.57889508150033776300144530938, 14.09302247404951393002372615696

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