Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.917 - 0.398i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + (0.199 − 2.99i)3-s + 2i·4-s + (4.37 + 2.41i)5-s + (3.19 − 2.79i)6-s + (4.76 + 5.12i)7-s + (−2 + 2i)8-s + (−8.92 − 1.19i)9-s + (1.95 + 6.79i)10-s + 6.70i·11-s + (5.98 + 0.398i)12-s + (16.0 − 16.0i)13-s + (−0.359 + 9.89i)14-s + (8.11 − 12.6i)15-s − 4·16-s + (−7.21 + 7.21i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.0664 − 0.997i)3-s + 0.5i·4-s + (0.875 + 0.483i)5-s + (0.532 − 0.465i)6-s + (0.680 + 0.732i)7-s + (−0.250 + 0.250i)8-s + (−0.991 − 0.132i)9-s + (0.195 + 0.679i)10-s + 0.609i·11-s + (0.498 + 0.0332i)12-s + (1.23 − 1.23i)13-s + (−0.0256 + 0.706i)14-s + (0.540 − 0.841i)15-s − 0.250·16-s + (−0.424 + 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.917 - 0.398i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.917 - 0.398i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.35959 + 0.489953i\)
\(L(\frac12)\)  \(\approx\)  \(2.35959 + 0.489953i\)
\(L(2)\)   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 \( 1 + (-1 - i)T \)
3 \( 1 + (-0.199 + 2.99i)T \)
5 \( 1 + (-4.37 - 2.41i)T \)
7 \( 1 + (-4.76 - 5.12i)T \)
good11 \( 1 - 6.70iT - 121T^{2} \)
13 \( 1 + (-16.0 + 16.0i)T - 169iT^{2} \)
17 \( 1 + (7.21 - 7.21i)T - 289iT^{2} \)
19 \( 1 - 8.06T + 361T^{2} \)
23 \( 1 + (-11.7 + 11.7i)T - 529iT^{2} \)
29 \( 1 - 6.17T + 841T^{2} \)
31 \( 1 + 41.4iT - 961T^{2} \)
37 \( 1 + (37.8 - 37.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 74.2T + 1.68e3T^{2} \)
43 \( 1 + (42.3 + 42.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (-39.4 + 39.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (44.4 - 44.4i)T - 2.80e3iT^{2} \)
59 \( 1 + 51.9iT - 3.48e3T^{2} \)
61 \( 1 - 15.0iT - 3.72e3T^{2} \)
67 \( 1 + (38.7 - 38.7i)T - 4.48e3iT^{2} \)
71 \( 1 + 128. iT - 5.04e3T^{2} \)
73 \( 1 + (54.2 - 54.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 25.7iT - 6.24e3T^{2} \)
83 \( 1 + (-27.7 - 27.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 32.5iT - 7.92e3T^{2} \)
97 \( 1 + (56.7 + 56.7i)T + 9.40e3iT^{2} \)
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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

−12.44278799795578878900868846238, −11.47177385841627021578097365640, −10.40605715482087025232212145858, −8.840734886553345752209642923974, −8.107609057231735136955665034441, −6.91057831086560845309748588132, −6.01298048794766947125884217246, −5.20311867062771621279371927833, −3.11762082102615599194149865824, −1.80523221726141990836538443545, 1.50965143082078983766904927878, 3.37704721907802563443498479324, 4.54355351376404580410407626065, 5.38631750637046158579747236100, 6.63330696874644365825363211849, 8.557264437891844712461487514471, 9.208441320375333025361747339870, 10.32295430196885279110624492370, 11.06024466356764328283906842042, 11.80549371782113552217145613463

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