Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.443 - 0.896i$
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 + (1.05 + 2.80i)3-s + (1.73 + i)4-s + (−1.47 + 4.77i)5-s + (−0.414 − 4.22i)6-s + (6.99 + 0.132i)7-s + (−1.99 − 2i)8-s + (−6.77 + 5.92i)9-s + (3.76 − 5.98i)10-s + (9.30 + 5.36i)11-s + (−0.979 + 5.91i)12-s + (−3.28 − 3.28i)13-s + (−9.51 − 2.74i)14-s + (−14.9 + 0.902i)15-s + (1.99 + 3.46i)16-s + (−16.0 + 4.29i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.351 + 0.936i)3-s + (0.433 + 0.250i)4-s + (−0.295 + 0.955i)5-s + (−0.0690 − 0.703i)6-s + (0.999 + 0.0189i)7-s + (−0.249 − 0.250i)8-s + (−0.752 + 0.658i)9-s + (0.376 − 0.598i)10-s + (0.845 + 0.488i)11-s + (−0.0816 + 0.493i)12-s + (−0.252 − 0.252i)13-s + (−0.679 − 0.195i)14-s + (−0.998 + 0.0601i)15-s + (0.124 + 0.216i)16-s + (−0.942 + 0.252i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.443 - 0.896i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.443 - 0.896i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.616307 + 0.992880i\)
\(L(\frac12)\)  \(\approx\)  \(0.616307 + 0.992880i\)
\(L(2)\)   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 \( 1 + (-1.05 - 2.80i)T \)
5 \( 1 + (1.47 - 4.77i)T \)
7 \( 1 + (-6.99 - 0.132i)T \)
good11 \( 1 + (-9.30 - 5.36i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.28 + 3.28i)T + 169iT^{2} \)
17 \( 1 + (16.0 - 4.29i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (1.04 + 1.81i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (3.40 - 12.7i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 21.8T + 841T^{2} \)
31 \( 1 + (-40.3 - 23.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (10.9 - 40.7i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 41.7T + 1.68e3T^{2} \)
43 \( 1 + (32.4 - 32.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-18.0 + 67.3i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (31.2 - 8.36i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (8.21 + 4.74i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-100. + 58.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.25 - 0.604i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 82.6iT - 5.04e3T^{2} \)
73 \( 1 + (-89.0 + 23.8i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (27.5 - 15.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-37.0 + 37.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (-136. + 78.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (41.8 - 41.8i)T - 9.40e3iT^{2} \)
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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

−11.91373656414047794697167116826, −11.26631100408514250661022148083, −10.48306690325144596426383047362, −9.614102168236460565043509128543, −8.560968233926274253827976319531, −7.69268032912329409407018703309, −6.49779576404805037157206781403, −4.79217150214538844147614282455, −3.59239757145501720130468352032, −2.15572126591673746211309706560, 0.805841323219734678829534011249, 2.11278398280029997100844934712, 4.24571776742608958763227770756, 5.75685847426773989402717383571, 6.97809023567063979557606967908, 7.998923480747264656382403921801, 8.667214660198602814156645908674, 9.405790580544237244022180704390, 11.14733974760935862953454197447, 11.74009257078057489176342549496

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