Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.690 + 0.723i$
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 + (3.25 − 3.79i)5-s + 2.44·6-s + (−2.26 − 6.62i)7-s + (1.99 − 2i)8-s + (2.59 + 1.50i)9-s + (3.05 − 6.37i)10-s + (−1.33 − 2.30i)11-s + (3.34 − 0.896i)12-s + (−9.16 + 9.16i)13-s + (−5.52 − 8.21i)14-s + (7.14 − 4.88i)15-s + (1.99 − 3.46i)16-s + (0.724 − 2.70i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.557 + 0.149i)3-s + (0.433 − 0.250i)4-s + (0.651 − 0.758i)5-s + 0.408·6-s + (−0.324 − 0.946i)7-s + (0.249 − 0.250i)8-s + (0.288 + 0.166i)9-s + (0.305 − 0.637i)10-s + (−0.121 − 0.209i)11-s + (0.278 − 0.0747i)12-s + (−0.704 + 0.704i)13-s + (−0.394 − 0.586i)14-s + (0.476 − 0.325i)15-s + (0.124 − 0.216i)16-s + (0.0426 − 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.690 + 0.723i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.690 + 0.723i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.59836 - 1.11282i\)
\(L(\frac12)\)  \(\approx\)  \(2.59836 - 1.11282i\)
\(L(2)\)   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 \( 1 + (-3.25 + 3.79i)T \)
7 \( 1 + (2.26 + 6.62i)T \)
good11 \( 1 + (1.33 + 2.30i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (9.16 - 9.16i)T - 169iT^{2} \)
17 \( 1 + (-0.724 + 2.70i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-16.5 - 9.57i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-9.36 - 34.9i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 12.4iT - 841T^{2} \)
31 \( 1 + (-5.33 - 9.24i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-21.9 + 5.88i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 1.17T + 1.68e3T^{2} \)
43 \( 1 + (-2.70 + 2.70i)T - 1.84e3iT^{2} \)
47 \( 1 + (71.7 - 19.2i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-7.83 - 2.09i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (93.6 - 54.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-35.0 + 60.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-4.51 + 16.8i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 66.2T + 5.04e3T^{2} \)
73 \( 1 + (34.9 + 9.37i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-83.6 - 48.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (83.4 - 83.4i)T - 6.88e3iT^{2} \)
89 \( 1 + (125. + 72.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (43.6 + 43.6i)T + 9.40e3iT^{2} \)
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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

−12.25337487320116621091551942213, −11.11448956210597084872168610505, −9.858969206405755337640736117260, −9.404246696440540546040290362213, −7.901296846562635591889954169560, −6.85741953904947639966345104522, −5.46691515301014609774521819154, −4.43010381291618910215919552799, −3.18231276796121196671395915725, −1.49092148665303647052783058366, 2.36809055144177624708369469435, 3.11385029831396190981097733133, 4.93439552849110193684678138822, 6.06011508829356483033455722099, 6.99789356725184528047330855073, 8.129706569916941411619152238517, 9.396636286537630179070321048535, 10.24156880071596277698558853651, 11.47586288752412439285850446561, 12.55760893858631848864518334148

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