Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.757 - 0.653i$
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 + (−3.99 + 3.00i)5-s + 2.44·6-s + (6.54 + 2.49i)7-s + (1.99 + 2i)8-s + (2.59 − 1.50i)9-s + (−6.55 + 2.64i)10-s + (1.59 − 2.75i)11-s + (3.34 + 0.896i)12-s + (16.7 + 16.7i)13-s + (8.02 + 5.79i)14-s + (−5.33 + 6.82i)15-s + (1.99 + 3.46i)16-s + (−3.89 − 14.5i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.557 − 0.149i)3-s + (0.433 + 0.250i)4-s + (−0.798 + 0.601i)5-s + 0.408·6-s + (0.934 + 0.356i)7-s + (0.249 + 0.250i)8-s + (0.288 − 0.166i)9-s + (−0.655 + 0.264i)10-s + (0.144 − 0.250i)11-s + (0.278 + 0.0747i)12-s + (1.28 + 1.28i)13-s + (0.573 + 0.414i)14-s + (−0.355 + 0.454i)15-s + (0.124 + 0.216i)16-s + (−0.229 − 0.855i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.757 - 0.653i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (193, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.757 - 0.653i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.48026 + 0.921884i\)
\(L(\frac12)\)  \(\approx\)  \(2.48026 + 0.921884i\)
\(L(2)\)   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 \( 1 + (3.99 - 3.00i)T \)
7 \( 1 + (-6.54 - 2.49i)T \)
good11 \( 1 + (-1.59 + 2.75i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-16.7 - 16.7i)T + 169iT^{2} \)
17 \( 1 + (3.89 + 14.5i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (13.5 - 7.79i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (1.45 - 5.41i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 47.4iT - 841T^{2} \)
31 \( 1 + (-0.731 + 1.26i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (53.3 + 14.3i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 27.2T + 1.68e3T^{2} \)
43 \( 1 + (16.8 + 16.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (-26.2 - 7.04i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-33.3 + 8.92i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (46.3 + 26.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-20.4 - 35.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (15.8 + 59.1i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 102.T + 5.04e3T^{2} \)
73 \( 1 + (-29.5 + 7.92i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-101. + 58.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (74.9 + 74.9i)T + 6.88e3iT^{2} \)
89 \( 1 + (-91.7 + 53.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (66.3 - 66.3i)T - 9.40e3iT^{2} \)
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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

−12.02599976411517844714875965544, −11.56871697049797807484473726821, −10.64501207255328367729743878341, −8.952334113193031043248897411074, −8.169878541879305880266667301420, −7.14173276773713026964808140022, −6.12751516410938256642019314483, −4.50638828894978447354490010729, −3.59861604507699894558294909858, −2.05138926077198614200135509175, 1.43035623140440470050321445640, 3.39672547529907491252577328023, 4.32958310498532622334679226214, 5.38808321472144553102805719143, 6.98671584768014575762924055375, 8.226532685967232841491399819782, 8.682073109160552251859114349330, 10.49675954138952857981541751448, 10.99813090083989424443594586664, 12.22297138019617359715379314376

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