Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.953 - 0.300i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.85 − 3.22i)2-s + (−2.70 + 1.29i)3-s + (−4.91 − 8.51i)4-s + (−4.36 + 2.43i)5-s + (−0.869 + 11.1i)6-s + (−4.87 − 5.02i)7-s − 21.7·8-s + (5.65 − 7.00i)9-s + (−0.272 + 18.5i)10-s + (10.0 − 5.82i)11-s + (24.3 + 16.7i)12-s − 9.22i·13-s + (−25.2 + 6.37i)14-s + (8.66 − 12.2i)15-s + (−20.7 + 35.8i)16-s + (1.56 + 2.70i)17-s + ⋯
L(s)  = 1  + (0.929 − 1.61i)2-s + (−0.902 + 0.431i)3-s + (−1.22 − 2.12i)4-s + (−0.873 + 0.487i)5-s + (−0.144 + 1.85i)6-s + (−0.696 − 0.717i)7-s − 2.71·8-s + (0.628 − 0.777i)9-s + (−0.0272 + 1.85i)10-s + (0.916 − 0.529i)11-s + (2.02 + 1.39i)12-s − 0.709i·13-s + (−1.80 + 0.455i)14-s + (0.577 − 0.816i)15-s + (−1.29 + 2.24i)16-s + (0.0919 + 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.953 - 0.300i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (44, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.953 - 0.300i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.148702 + 0.965312i\)
\(L(\frac12)\)  \(\approx\)  \(0.148702 + 0.965312i\)
\(L(2)\)   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 \( 1 + (2.70 - 1.29i)T \)
5 \( 1 + (4.36 - 2.43i)T \)
7 \( 1 + (4.87 + 5.02i)T \)
good2 \( 1 + (-1.85 + 3.22i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (-10.0 + 5.82i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 9.22iT - 169T^{2} \)
17 \( 1 + (-1.56 - 2.70i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.39 + 9.34i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (2.93 - 5.08i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 38.3iT - 841T^{2} \)
31 \( 1 + (15.7 + 27.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (20.0 + 11.5i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 22.7iT - 1.68e3T^{2} \)
43 \( 1 + 29.1iT - 1.84e3T^{2} \)
47 \( 1 + (-30.0 + 52.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-27.9 - 48.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-23.2 + 13.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (19.8 - 34.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-86.4 + 49.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 62.5iT - 5.04e3T^{2} \)
73 \( 1 + (-37.5 + 21.6i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-15.5 + 26.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 93.5T + 6.88e3T^{2} \)
89 \( 1 + (34.2 + 19.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 119. iT - 9.40e3T^{2} \)
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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

−12.57881647288034857843360884194, −11.85903492734455754762680212922, −10.92493469577396481952361169439, −10.43160234305585390386038920885, −9.245766564844852759354007588420, −6.89028809415216311446988549476, −5.51977209093637827140673443843, −4.04529476647648938905170817977, −3.39662783558458804280174924995, −0.62788813843130635130953740108, 3.95138980950879041366906497634, 5.05345128363409968311684225350, 6.24249054057629969230936734746, 7.02469518141656898301112494998, 8.138088184606921939551383007964, 9.384273760510351464414524736777, 11.70302602465683668901876685458, 12.26040698732709398950413299147, 13.02218777988649115660262994584, 14.21298579207547998031041588059

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