Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.641 + 0.766i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)2-s + (−1.5 + 0.866i)3-s + (−0.999 − 1.73i)4-s + (−1.93 − 1.11i)5-s + 2.44i·6-s + (6.99 + 0.325i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (−2.73 + 1.58i)10-s + (−6.09 − 10.5i)11-s + (2.99 + 1.73i)12-s − 25.3i·13-s + (5.34 − 8.33i)14-s + 3.87·15-s + (−2.00 + 3.46i)16-s + (−24.9 + 14.3i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + 0.408i·6-s + (0.998 + 0.0465i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.273 + 0.158i)10-s + (−0.554 − 0.959i)11-s + (0.249 + 0.144i)12-s − 1.95i·13-s + (0.381 − 0.595i)14-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (−1.46 + 0.846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.641 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.641 + 0.766i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (61, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.641 + 0.766i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.501285 - 1.07325i\)
\(L(\frac12)\)  \(\approx\)  \(0.501285 - 1.07325i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (-6.99 - 0.325i)T \)
good11 \( 1 + (6.09 + 10.5i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 25.3iT - 169T^{2} \)
17 \( 1 + (24.9 - 14.3i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (13.9 + 8.03i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-11.8 + 20.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 27.9T + 841T^{2} \)
31 \( 1 + (-20.0 + 11.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-14.5 + 25.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 56.9iT - 1.68e3T^{2} \)
43 \( 1 - 7.83T + 1.84e3T^{2} \)
47 \( 1 + (19.7 + 11.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (24.2 + 42.0i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (62.3 - 36.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-99.2 - 57.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-35.2 - 61.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 6.41T + 5.04e3T^{2} \)
73 \( 1 + (34.2 - 19.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-27.3 + 47.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 135. iT - 6.88e3T^{2} \)
89 \( 1 + (-124. - 72.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 78.9iT - 9.40e3T^{2} \)
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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

−11.60476379485093850196795032161, −10.83539515463937403650477710305, −10.41002538342306556985558978455, −8.670958991607291102391784231925, −8.081022224542557614984889853938, −6.28005925240611030521269822620, −5.17955675272123182621723368833, −4.29952963184732019128916221939, −2.73475786541821102652990224151, −0.63310162333397786645667231835, 2.08495543081594910551898217251, 4.35914496735298562416437316675, 4.90821505543221057463820636901, 6.57360898420384065273245506829, 7.16417052608293016487699763929, 8.262232679668601879245646892311, 9.384777887579909696233446340640, 10.88307745729664457465107549250, 11.61869328563836834977618798565, 12.39428697860764764647896458182

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