Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.999 + 0.0140i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.773 − 2.88i)2-s + (0.448 − 1.67i)3-s + (−4.27 + 2.46i)4-s + (4.24 − 2.63i)5-s − 5.17·6-s + (−4.68 − 5.19i)7-s + (1.96 + 1.96i)8-s + (−2.59 − 1.50i)9-s + (−10.9 − 10.2i)10-s + (6.71 + 11.6i)11-s + (2.21 + 8.24i)12-s + (−3.94 − 3.94i)13-s + (−11.3 + 17.5i)14-s + (−2.50 − 8.28i)15-s + (−5.70 + 9.88i)16-s + (9.46 + 2.53i)17-s + ⋯
L(s)  = 1  + (−0.386 − 1.44i)2-s + (0.149 − 0.557i)3-s + (−1.06 + 0.616i)4-s + (0.849 − 0.527i)5-s − 0.862·6-s + (−0.669 − 0.742i)7-s + (0.245 + 0.245i)8-s + (−0.288 − 0.166i)9-s + (−1.09 − 1.02i)10-s + (0.610 + 1.05i)11-s + (0.184 + 0.687i)12-s + (−0.303 − 0.303i)13-s + (−0.812 + 1.25i)14-s + (−0.167 − 0.552i)15-s + (−0.356 + 0.617i)16-s + (0.556 + 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0140i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.999 + 0.0140i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (58, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.999 + 0.0140i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.00803240 - 1.14539i\)
\(L(\frac12)\)  \(\approx\)  \(0.00803240 - 1.14539i\)
\(L(2)\)   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 \( 1 + (-0.448 + 1.67i)T \)
5 \( 1 + (-4.24 + 2.63i)T \)
7 \( 1 + (4.68 + 5.19i)T \)
good2 \( 1 + (0.773 + 2.88i)T + (-3.46 + 2i)T^{2} \)
11 \( 1 + (-6.71 - 11.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (3.94 + 3.94i)T + 169iT^{2} \)
17 \( 1 + (-9.46 - 2.53i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (-4.57 - 2.64i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-7.39 + 1.98i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 36.7iT - 841T^{2} \)
31 \( 1 + (9.71 + 16.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (11.7 + 43.7i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 57.9T + 1.68e3T^{2} \)
43 \( 1 + (-46.3 - 46.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (-16.4 - 61.3i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-10.4 + 38.8i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-60.4 + 34.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (39.4 - 68.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (107. + 28.7i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 121.T + 5.04e3T^{2} \)
73 \( 1 + (1.25 - 4.70i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-116. - 66.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-99.6 - 99.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (-20.5 - 11.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (40.9 - 40.9i)T - 9.40e3iT^{2} \)
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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

−12.73105339801808646682968470383, −12.17766359519681682390801630364, −10.78116562509320838318768487710, −9.727780291191315969032308490247, −9.278076235115301562933729221472, −7.57816589142645120686705603089, −6.13873383709995750414573562037, −4.14446137680346193477358866151, −2.46758315819647540987043693026, −1.03217387953203230478215417484, 3.06360997999004039262142315402, 5.34962449832859638423304633242, 6.14261227494202340985560563289, 7.18914643783783469910460874409, 8.842932363219874622096445408930, 9.243819021626707292952745851992, 10.49293677774619481739704588245, 11.93337801484094567538888952064, 13.56818194767915130814920315996, 14.37682907088573475298507676085

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