Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.923 + 0.384i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (0.941 − 2.84i)3-s + (1.73 + i)4-s + (2.72 − 4.19i)5-s + (−2.32 + 3.54i)6-s + (−5.35 − 4.50i)7-s + (−1.99 − 2i)8-s + (−7.22 − 5.36i)9-s + (−5.25 + 4.73i)10-s + (−8.15 − 4.70i)11-s + (4.47 − 3.99i)12-s + (16.1 + 16.1i)13-s + (5.66 + 8.11i)14-s + (−9.37 − 11.7i)15-s + (1.99 + 3.46i)16-s + (9.03 − 2.42i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.313 − 0.949i)3-s + (0.433 + 0.250i)4-s + (0.544 − 0.838i)5-s + (−0.388 + 0.591i)6-s + (−0.765 − 0.643i)7-s + (−0.249 − 0.250i)8-s + (−0.802 − 0.596i)9-s + (−0.525 + 0.473i)10-s + (−0.741 − 0.428i)11-s + (0.373 − 0.332i)12-s + (1.24 + 1.24i)13-s + (0.404 + 0.579i)14-s + (−0.625 − 0.780i)15-s + (0.124 + 0.216i)16-s + (0.531 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.923 + 0.384i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.923 + 0.384i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.195457 - 0.977742i\)
\(L(\frac12)\)  \(\approx\)  \(0.195457 - 0.977742i\)
\(L(2)\)   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 \( 1 + (-0.941 + 2.84i)T \)
5 \( 1 + (-2.72 + 4.19i)T \)
7 \( 1 + (5.35 + 4.50i)T \)
good11 \( 1 + (8.15 + 4.70i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-16.1 - 16.1i)T + 169iT^{2} \)
17 \( 1 + (-9.03 + 2.42i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (13.1 + 22.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (6.16 - 23.0i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 34.0T + 841T^{2} \)
31 \( 1 + (-2.88 - 1.66i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-12.3 + 46.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 3.31T + 1.68e3T^{2} \)
43 \( 1 + (-28.9 + 28.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (0.830 - 3.09i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-56.7 + 15.2i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (33.5 + 19.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-80.5 + 46.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-3.66 + 0.981i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 26.4iT - 5.04e3T^{2} \)
73 \( 1 + (-27.3 + 7.31i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-38.5 + 22.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-70.6 + 70.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (-118. + 68.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (118. - 118. i)T - 9.40e3iT^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.69565928157137964891422817945, −10.76300035449772601062528626858, −9.386021832062671068772088846944, −8.894990425538976108695195622127, −7.76890349725558925880566380653, −6.72900017778759086321925035660, −5.74564343146975406093934841676, −3.71467137356730193521489017424, −2.07032137862268001702652954576, −0.65175778701799644335082374552, 2.46235333114679471334553395141, 3.53986317184568405502947946633, 5.58230804382563044656531421432, 6.21234725135039440698773888767, 7.83899869446597024615073571629, 8.673576779846331603217794568909, 9.895611092611327135120486823525, 10.27760277715211181034766817784, 11.08423953053104800757913748283, 12.58211145086673036931095430991

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