Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.000795 - 0.999i$
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 + (−0.438 + 4.98i)5-s + 2.44·6-s + (6.08 − 3.46i)7-s + (−1.99 + 2i)8-s + (2.59 + 1.50i)9-s + (−1.22 − 6.96i)10-s + (−5.62 − 9.74i)11-s + (−3.34 + 0.896i)12-s + (−5.84 + 5.84i)13-s + (−7.04 + 6.95i)14-s + (2.96 − 8.13i)15-s + (1.99 − 3.46i)16-s + (−6.65 + 24.8i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.557 − 0.149i)3-s + (0.433 − 0.250i)4-s + (−0.0876 + 0.996i)5-s + 0.408·6-s + (0.869 − 0.494i)7-s + (−0.249 + 0.250i)8-s + (0.288 + 0.166i)9-s + (−0.122 − 0.696i)10-s + (−0.511 − 0.885i)11-s + (−0.278 + 0.0747i)12-s + (−0.449 + 0.449i)13-s + (−0.503 + 0.496i)14-s + (0.197 − 0.542i)15-s + (0.124 − 0.216i)16-s + (−0.391 + 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.000795 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.000795 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.000795 - 0.999i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.000795 - 0.999i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.577138 + 0.576679i\)
\(L(\frac12)\)  \(\approx\)  \(0.577138 + 0.576679i\)
\(L(2)\)   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 \( 1 + (0.438 - 4.98i)T \)
7 \( 1 + (-6.08 + 3.46i)T \)
good11 \( 1 + (5.62 + 9.74i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (5.84 - 5.84i)T - 169iT^{2} \)
17 \( 1 + (6.65 - 24.8i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-27.5 - 15.8i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-10.6 - 39.8i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 8.29iT - 841T^{2} \)
31 \( 1 + (-6.00 - 10.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (23.6 - 6.32i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 42.9T + 1.68e3T^{2} \)
43 \( 1 + (37.4 - 37.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (35.8 - 9.60i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-11.0 - 2.96i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (52.5 - 30.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-18.8 + 32.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-21.1 + 78.8i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 1.19T + 5.04e3T^{2} \)
73 \( 1 + (-69.0 - 18.5i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-71.0 - 41.0i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-103. + 103. i)T - 6.88e3iT^{2} \)
89 \( 1 + (-12.6 - 7.30i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-24.6 - 24.6i)T + 9.40e3iT^{2} \)
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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

−11.96425760009372092209891637014, −11.15827573730483144647428325120, −10.63248450828343708424905223688, −9.612163078698871145206732501791, −8.079922475366608108583738099009, −7.47947983029531616194397097387, −6.35907176357222955407369165418, −5.27586439062434398652235293522, −3.47661057575278607481415288791, −1.56102405399834887693312593588, 0.64379541355079312629408121681, 2.42590544698280049502159425851, 4.73479403818365116502087385845, 5.25042007754116411493660939050, 7.03328309671912057013895291480, 7.974532116535570324712861944670, 9.059977689174714893606162850141, 9.812486262643699871207889972864, 10.99620646083806705920373067637, 11.88468971052941919432339665499

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