Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.710 + 0.704i$
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 + (−1.25 − 4.83i)5-s + 2.44·6-s + (5.72 + 4.03i)7-s + (−1.99 + 2i)8-s + (2.59 + 1.50i)9-s + (3.49 + 6.14i)10-s + (−5.44 − 9.42i)11-s + (−3.34 + 0.896i)12-s + (−4.13 + 4.13i)13-s + (−9.29 − 3.41i)14-s + (−0.0638 + 8.66i)15-s + (1.99 − 3.46i)16-s + (−0.489 + 1.82i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.557 − 0.149i)3-s + (0.433 − 0.250i)4-s + (−0.251 − 0.967i)5-s + 0.408·6-s + (0.817 + 0.576i)7-s + (−0.249 + 0.250i)8-s + (0.288 + 0.166i)9-s + (0.349 + 0.614i)10-s + (−0.494 − 0.856i)11-s + (−0.278 + 0.0747i)12-s + (−0.318 + 0.318i)13-s + (−0.663 − 0.243i)14-s + (−0.00425 + 0.577i)15-s + (0.124 − 0.216i)16-s + (−0.0287 + 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.704i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.710 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.710 + 0.704i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.710 + 0.704i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.183486 - 0.445682i\)
\(L(\frac12)\)  \(\approx\)  \(0.183486 - 0.445682i\)
\(L(2)\)   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 \( 1 + (1.25 + 4.83i)T \)
7 \( 1 + (-5.72 - 4.03i)T \)
good11 \( 1 + (5.44 + 9.42i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (4.13 - 4.13i)T - 169iT^{2} \)
17 \( 1 + (0.489 - 1.82i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (27.1 + 15.6i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (5.85 + 21.8i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 35.4iT - 841T^{2} \)
31 \( 1 + (7.10 + 12.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.4 + 3.60i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 75.8T + 1.68e3T^{2} \)
43 \( 1 + (5.54 - 5.54i)T - 1.84e3iT^{2} \)
47 \( 1 + (89.5 - 23.9i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-73.5 - 19.6i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-39.8 + 22.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-46.7 + 80.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (6.47 - 24.1i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 4.10T + 5.04e3T^{2} \)
73 \( 1 + (-94.5 - 25.3i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-25.8 - 14.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (42.2 - 42.2i)T - 6.88e3iT^{2} \)
89 \( 1 + (-71.6 - 41.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (60.0 + 60.0i)T + 9.40e3iT^{2} \)
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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

−11.59281811381526970739662618161, −10.97963091500034513435485856453, −9.719059630896920821144108599336, −8.466050120427528862224252019622, −8.175566201568307102309115545797, −6.62189250856174654431245403315, −5.49644910826214341003396640640, −4.49870060241131833515333082480, −2.09977682576911866484330384556, −0.34628223369835469488790338967, 1.92188620600466867750849031218, 3.70331199448241574600702504152, 5.12294960957611749013609946323, 6.67424869308350305345351582778, 7.45283854383703956350936311060, 8.398000466120613908937145949078, 10.08949001651813688900756321297, 10.39594230993088967856028251692, 11.34118711787347606623355003985, 12.14101480940740019178317905073

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