Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.672 + 0.739i$
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 + (−1.67 − 0.448i)3-s + (1.73 − i)4-s + (−4.95 + 0.641i)5-s − 2.44·6-s + (2.95 − 6.34i)7-s + (1.99 − 2i)8-s + (2.59 + 1.50i)9-s + (−6.53 + 2.69i)10-s + (−9.98 − 17.2i)11-s + (−3.34 + 0.896i)12-s + (−13.3 + 13.3i)13-s + (1.71 − 9.74i)14-s + (8.58 + 1.14i)15-s + (1.99 − 3.46i)16-s + (5.85 − 21.8i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.557 − 0.149i)3-s + (0.433 − 0.250i)4-s + (−0.991 + 0.128i)5-s − 0.408·6-s + (0.422 − 0.906i)7-s + (0.249 − 0.250i)8-s + (0.288 + 0.166i)9-s + (−0.653 + 0.269i)10-s + (−0.907 − 1.57i)11-s + (−0.278 + 0.0747i)12-s + (−1.02 + 1.02i)13-s + (0.122 − 0.696i)14-s + (0.572 + 0.0765i)15-s + (0.124 − 0.216i)16-s + (0.344 − 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.739i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.672 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.672 + 0.739i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.672 + 0.739i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.454459 - 1.02739i\)
\(L(\frac12)\)  \(\approx\)  \(0.454459 - 1.02739i\)
\(L(2)\)   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 \( 1 + (1.67 + 0.448i)T \)
5 \( 1 + (4.95 - 0.641i)T \)
7 \( 1 + (-2.95 + 6.34i)T \)
good11 \( 1 + (9.98 + 17.2i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (13.3 - 13.3i)T - 169iT^{2} \)
17 \( 1 + (-5.85 + 21.8i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (14.8 + 8.58i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (5.43 + 20.2i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 23.1iT - 841T^{2} \)
31 \( 1 + (-27.7 - 48.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-22.2 + 5.97i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 21.6T + 1.68e3T^{2} \)
43 \( 1 + (-14.6 + 14.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-73.1 + 19.6i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-4.01 - 1.07i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (26.7 - 15.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (1.75 - 3.04i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-0.764 + 2.85i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 11.8T + 5.04e3T^{2} \)
73 \( 1 + (-21.0 - 5.65i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-65.5 - 37.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-28.2 + 28.2i)T - 6.88e3iT^{2} \)
89 \( 1 + (124. + 71.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-56.0 - 56.0i)T + 9.40e3iT^{2} \)
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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

−11.76528878912712440477167064657, −11.00885160543671417943813691794, −10.37921725797669735774501725276, −8.614280453727753569690355423366, −7.43729502318055857294851132442, −6.70707802309942805351448147943, −5.13741334263275157924698697315, −4.33224625853716547702958289744, −2.88585994167144470027767108206, −0.52507656120272725444614665177, 2.38348763468158809267363592519, 4.15270699139456003635967349431, 5.03679041458380092353542563790, 6.02711419393269882905555636113, 7.62255963337145432029862053992, 8.016826198676991060876129434260, 9.793589258250209000812471398138, 10.72958331179610831124996465308, 11.90064158258300163668480418403, 12.40080019971682768377170903735

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