Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.976 + 0.215i$
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 + (2.17 + 4.50i)5-s − 2.44·6-s + (1.63 − 6.80i)7-s + (1.99 − 2i)8-s + (2.59 + 1.50i)9-s + (4.61 + 5.35i)10-s + (9.66 + 16.7i)11-s + (−3.34 + 0.896i)12-s + (10.3 − 10.3i)13-s + (−0.253 − 9.89i)14-s + (−1.61 − 8.50i)15-s + (1.99 − 3.46i)16-s + (6.19 − 23.1i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.557 − 0.149i)3-s + (0.433 − 0.250i)4-s + (0.434 + 0.900i)5-s − 0.408·6-s + (0.234 − 0.972i)7-s + (0.249 − 0.250i)8-s + (0.288 + 0.166i)9-s + (0.461 + 0.535i)10-s + (0.879 + 1.52i)11-s + (−0.278 + 0.0747i)12-s + (0.799 − 0.799i)13-s + (−0.0180 − 0.706i)14-s + (−0.107 − 0.567i)15-s + (0.124 − 0.216i)16-s + (0.364 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.976 + 0.215i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.976 + 0.215i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.21908 - 0.241987i\)
\(L(\frac12)\)  \(\approx\)  \(2.21908 - 0.241987i\)
\(L(2)\)   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 \( 1 + (-2.17 - 4.50i)T \)
7 \( 1 + (-1.63 + 6.80i)T \)
good11 \( 1 + (-9.66 - 16.7i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.3 + 10.3i)T - 169iT^{2} \)
17 \( 1 + (-6.19 + 23.1i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-2.70 - 1.55i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-9.18 - 34.2i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 36.5iT - 841T^{2} \)
31 \( 1 + (1.49 + 2.59i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-0.422 + 0.113i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 64.3T + 1.68e3T^{2} \)
43 \( 1 + (38.9 - 38.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (-10.7 + 2.88i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-29.6 - 7.95i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (34.4 - 19.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (21.4 - 37.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (1.06 - 3.98i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 81.1T + 5.04e3T^{2} \)
73 \( 1 + (101. + 27.1i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (25.8 + 14.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-6.76 + 6.76i)T - 6.88e3iT^{2} \)
89 \( 1 + (-101. - 58.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (63.8 + 63.8i)T + 9.40e3iT^{2} \)
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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

−11.90252417635360158953008750618, −11.35954758941031122060454801176, −10.25384082768681216131246885843, −9.676005544276715648136170510351, −7.50088232697279452377841890898, −6.97330762077241383771217527957, −5.82587170910145519450461797943, −4.59947854525627351092200813267, −3.31843016575288871376800450016, −1.51519254987444798371562829155, 1.52150681072174324212917559406, 3.59064327675062804067085160368, 4.89031075400064069864832466870, 5.90473105523534661910840984625, 6.46907266443963423831496747492, 8.640097533089248202577942447028, 8.743805376129813587754518098552, 10.47608827708834975924964594950, 11.49436422657671126699630087755, 12.18653621149923408534148024617

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