Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.812 + 0.583i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (2.42 + 1.76i)3-s + (1.73 + i)4-s + (−4.98 + 0.322i)5-s + (−2.67 − 3.29i)6-s + (−4.85 − 5.04i)7-s + (−1.99 − 2i)8-s + (2.80 + 8.55i)9-s + (6.93 + 1.38i)10-s + (−17.0 − 9.81i)11-s + (2.44 + 5.47i)12-s + (−7.73 − 7.73i)13-s + (4.78 + 8.66i)14-s + (−12.6 − 8.00i)15-s + (1.99 + 3.46i)16-s + (12.2 − 3.28i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.809 + 0.586i)3-s + (0.433 + 0.250i)4-s + (−0.997 + 0.0645i)5-s + (−0.445 − 0.549i)6-s + (−0.693 − 0.720i)7-s + (−0.249 − 0.250i)8-s + (0.311 + 0.950i)9-s + (0.693 + 0.138i)10-s + (−1.54 − 0.892i)11-s + (0.203 + 0.456i)12-s + (−0.594 − 0.594i)13-s + (0.342 + 0.618i)14-s + (−0.845 − 0.533i)15-s + (0.124 + 0.216i)16-s + (0.721 − 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.812 + 0.583i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.812 + 0.583i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0862758 - 0.268054i\)
\(L(\frac12)\)  \(\approx\)  \(0.0862758 - 0.268054i\)
\(L(2)\)   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 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 + (-2.42 - 1.76i)T \)
5 \( 1 + (4.98 - 0.322i)T \)
7 \( 1 + (4.85 + 5.04i)T \)
good11 \( 1 + (17.0 + 9.81i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (7.73 + 7.73i)T + 169iT^{2} \)
17 \( 1 + (-12.2 + 3.28i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-0.306 - 0.530i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.26 + 12.1i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 29.8T + 841T^{2} \)
31 \( 1 + (17.4 + 10.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (3.54 - 13.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 75.0T + 1.68e3T^{2} \)
43 \( 1 + (46.7 - 46.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-13.4 + 50.2i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-64.3 + 17.2i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-64.0 - 36.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (19.2 - 11.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-8.82 + 2.36i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 4.20iT - 5.04e3T^{2} \)
73 \( 1 + (-22.9 + 6.15i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (67.0 - 38.7i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-41.1 + 41.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (1.42 - 0.825i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-92.5 + 92.5i)T - 9.40e3iT^{2} \)
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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

−11.45655988065601123907450564814, −10.42426084176676126631120226184, −10.03160637403033775534477260097, −8.636620088824325845176459712065, −7.896679653050133447073804548956, −7.17740874246573032054742340345, −5.21499772303879746632791191730, −3.63620188195650305948858947588, −2.86162845760337218409598846730, −0.16571548572954257938385230945, 2.16259830272827244927663764025, 3.43174351456315598978755130130, 5.28891211340299707765424012900, 6.94730356438033307615603116164, 7.55380175488423997436422992268, 8.453307482432435611515632905117, 9.406742411722340435006682102578, 10.30137025435911345380727892922, 11.78572779297559950638810051316, 12.45649032088499146557358825269

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