Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.850 + 0.526i$
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 + (−2.99 − 0.157i)3-s + (1.73 + i)4-s + (2.05 + 4.55i)5-s + (4.03 + 1.31i)6-s + (−6.80 − 1.63i)7-s + (−1.99 − 2i)8-s + (8.95 + 0.945i)9-s + (−1.14 − 6.97i)10-s + (7.66 + 4.42i)11-s + (−5.03 − 3.26i)12-s + (−5.26 − 5.26i)13-s + (8.69 + 4.72i)14-s + (−5.44 − 13.9i)15-s + (1.99 + 3.46i)16-s + (−9.09 + 2.43i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.998 − 0.0525i)3-s + (0.433 + 0.250i)4-s + (0.411 + 0.911i)5-s + (0.672 + 0.218i)6-s + (−0.972 − 0.233i)7-s + (−0.249 − 0.250i)8-s + (0.994 + 0.105i)9-s + (−0.114 − 0.697i)10-s + (0.696 + 0.402i)11-s + (−0.419 − 0.272i)12-s + (−0.405 − 0.405i)13-s + (0.621 + 0.337i)14-s + (−0.363 − 0.931i)15-s + (0.124 + 0.216i)16-s + (−0.534 + 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.850 + 0.526i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.850 + 0.526i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0429669 - 0.151003i\)
\(L(\frac12)\)  \(\approx\)  \(0.0429669 - 0.151003i\)
\(L(2)\)   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 \( 1 + (2.99 + 0.157i)T \)
5 \( 1 + (-2.05 - 4.55i)T \)
7 \( 1 + (6.80 + 1.63i)T \)
good11 \( 1 + (-7.66 - 4.42i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (5.26 + 5.26i)T + 169iT^{2} \)
17 \( 1 + (9.09 - 2.43i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (16.3 + 28.3i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.23 + 12.0i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 48.1T + 841T^{2} \)
31 \( 1 + (42.0 + 24.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-4.02 + 15.0i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 45.6T + 1.68e3T^{2} \)
43 \( 1 + (17.0 - 17.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (-18.2 + 67.9i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-23.5 + 6.32i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (31.3 + 18.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-41.5 + 23.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-20.6 + 5.53i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 42.3iT - 5.04e3T^{2} \)
73 \( 1 + (118. - 31.8i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-53.6 + 31.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (113. - 113. i)T - 6.88e3iT^{2} \)
89 \( 1 + (104. - 60.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-72.8 + 72.8i)T - 9.40e3iT^{2} \)
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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

−11.34835216039724589281916205100, −10.83520429069014610121064286938, −9.870056065498766409527723361237, −9.169023858934232292107647857670, −7.23676567200838977724701317706, −6.80099158452437556915664027616, −5.76558670964054706795916669794, −3.97580870467100602537368117719, −2.27541392127793534040597389036, −0.12038259933104606863489759727, 1.62816109637419878205495961299, 4.03581415688882346017761851071, 5.59288179753727820633327210147, 6.23327669659264766238652882641, 7.38054487373436891677815477312, 8.928899368695613044945961770819, 9.490878807904127896409220964051, 10.46455695846715025831550817764, 11.55792219475300213610947074755, 12.45871948184560084419391573059

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