Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.900 + 0.434i$
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 + (−1.22 − 4.84i)5-s + 2.44·6-s + (6.32 − 3.00i)7-s + (1.99 + 2i)8-s + (2.59 − 1.50i)9-s + (0.0950 − 7.07i)10-s + (−1.95 + 3.38i)11-s + (3.34 + 0.896i)12-s + (−8.93 − 8.93i)13-s + (9.73 − 1.78i)14-s + (−4.22 − 7.55i)15-s + (1.99 + 3.46i)16-s + (3.83 + 14.3i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.557 − 0.149i)3-s + (0.433 + 0.250i)4-s + (−0.245 − 0.969i)5-s + 0.408·6-s + (0.903 − 0.428i)7-s + (0.249 + 0.250i)8-s + (0.288 − 0.166i)9-s + (0.00950 − 0.707i)10-s + (−0.177 + 0.308i)11-s + (0.278 + 0.0747i)12-s + (−0.687 − 0.687i)13-s + (0.695 − 0.127i)14-s + (−0.281 − 0.503i)15-s + (0.124 + 0.216i)16-s + (0.225 + 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.900 + 0.434i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (193, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.900 + 0.434i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.68521 - 0.613427i\)
\(L(\frac12)\)  \(\approx\)  \(2.68521 - 0.613427i\)
\(L(2)\)   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 \( 1 + (1.22 + 4.84i)T \)
7 \( 1 + (-6.32 + 3.00i)T \)
good11 \( 1 + (1.95 - 3.38i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (8.93 + 8.93i)T + 169iT^{2} \)
17 \( 1 + (-3.83 - 14.3i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-28.0 + 16.1i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (3.30 - 12.3i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 26.6iT - 841T^{2} \)
31 \( 1 + (17.0 - 29.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (6.25 + 1.67i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 26.0T + 1.68e3T^{2} \)
43 \( 1 + (-21.0 - 21.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (40.1 + 10.7i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (43.7 - 11.7i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-20.7 - 12.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-8.39 - 14.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (15.2 + 57.0i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 132.T + 5.04e3T^{2} \)
73 \( 1 + (8.56 - 2.29i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (90.6 - 52.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-94.1 - 94.1i)T + 6.88e3iT^{2} \)
89 \( 1 + (-43.2 + 24.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-118. + 118. i)T - 9.40e3iT^{2} \)
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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

−12.31823581349843533573413750446, −11.37299313519081354697020857799, −10.13060282665813179975457920065, −8.887892029652440680393054222065, −7.86683570309964470157477114537, −7.24604282273214903868688617005, −5.39895129467441313915611561416, −4.68237413282858707235908177181, −3.32331447838074438764018564544, −1.49301966346859662059372150149, 2.14713641227221779299109045059, 3.28156144074794620439978518495, 4.58929901889668377729438572224, 5.80870188476874596154668924394, 7.23699180415655097091937957683, 7.955807657640525244506074141281, 9.426250788086561904062461575992, 10.36097502244867056071783395595, 11.59222723501077721846475946243, 11.87364585689834868650512311268

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