Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.888 - 0.459i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 1.36i)2-s + (−2.99 − 0.145i)3-s + (−1.73 + i)4-s + (3.20 + 3.83i)5-s + (−0.898 − 4.14i)6-s + (6.79 + 1.68i)7-s + (−2 − 1.99i)8-s + (8.95 + 0.871i)9-s + (−4.06 + 5.78i)10-s + (−13.7 + 7.94i)11-s + (5.33 − 2.74i)12-s + (−5.32 − 5.32i)13-s + (0.185 + 9.89i)14-s + (−9.05 − 11.9i)15-s + (1.99 − 3.46i)16-s + (−4.55 + 16.9i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.998 − 0.0484i)3-s + (−0.433 + 0.250i)4-s + (0.641 + 0.767i)5-s + (−0.149 − 0.691i)6-s + (0.970 + 0.240i)7-s + (−0.250 − 0.249i)8-s + (0.995 + 0.0968i)9-s + (−0.406 + 0.578i)10-s + (−1.25 + 0.722i)11-s + (0.444 − 0.228i)12-s + (−0.409 − 0.409i)13-s + (0.0132 + 0.706i)14-s + (−0.603 − 0.797i)15-s + (0.124 − 0.216i)16-s + (−0.267 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.888 - 0.459i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (47, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.888 - 0.459i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.249855 + 1.02713i\)
\(L(\frac12)\)  \(\approx\)  \(0.249855 + 1.02713i\)
\(L(2)\)   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 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 + (2.99 + 0.145i)T \)
5 \( 1 + (-3.20 - 3.83i)T \)
7 \( 1 + (-6.79 - 1.68i)T \)
good11 \( 1 + (13.7 - 7.94i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (5.32 + 5.32i)T + 169iT^{2} \)
17 \( 1 + (4.55 - 16.9i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (5.62 - 9.74i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (39.1 - 10.4i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 22.8T + 841T^{2} \)
31 \( 1 + (-36.0 + 20.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (28.8 - 7.72i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 55.0T + 1.68e3T^{2} \)
43 \( 1 + (14.2 - 14.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-25.2 + 6.77i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-15.8 + 59.0i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (10.7 - 6.18i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-14.5 - 8.40i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (27.8 - 103. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 34.7iT - 5.04e3T^{2} \)
73 \( 1 + (15.2 - 56.8i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (40.5 + 23.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-75.6 + 75.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (-116. - 67.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-75.7 + 75.7i)T - 9.40e3iT^{2} \)
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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

−12.58604237125067451451114842943, −11.64322793084274314299858984234, −10.39353336159090341270420584574, −10.04699568651010040901377603773, −8.181118825708224972118520793372, −7.40655136231768313079359482309, −6.16735341576426326043366504434, −5.46576565171686159917535220534, −4.36245757491806561221845444797, −2.13031198786921479351296502780, 0.62450880245902650615264567236, 2.24054750204520729683568384954, 4.52378663048864866343011670613, 5.07171523565584756456251103651, 6.17101655884412385326901536084, 7.77739255161619807994719742584, 8.936871646770619780815085521757, 10.17121787682983779393874465639, 10.74696764905785837329115024565, 11.82026028975446046500120195222

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