Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.688 - 0.725i$
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 + (−1.22 − 1.22i)3-s − 2i·4-s + (−4.32 + 2.51i)5-s − 2.44·6-s + (−1.87 + 1.87i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−1.80 + 6.83i)10-s − 14.0·11-s + (−2.44 + 2.44i)12-s + (−6.03 − 6.03i)13-s + 3.74i·14-s + (8.37 + 2.20i)15-s − 4·16-s + (−9.54 + 9.54i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (−0.864 + 0.503i)5-s − 0.408·6-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.180 + 0.683i)10-s − 1.28·11-s + (−0.204 + 0.204i)12-s + (−0.464 − 0.464i)13-s + 0.267i·14-s + (0.558 + 0.147i)15-s − 0.250·16-s + (−0.561 + 0.561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.688 - 0.725i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.688 - 0.725i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.00231657 + 0.00539281i\)
\(L(\frac12)\)  \(\approx\)  \(0.00231657 + 0.00539281i\)
\(L(2)\)   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 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (4.32 - 2.51i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good11 \( 1 + 14.0T + 121T^{2} \)
13 \( 1 + (6.03 + 6.03i)T + 169iT^{2} \)
17 \( 1 + (9.54 - 9.54i)T - 289iT^{2} \)
19 \( 1 - 21.6iT - 361T^{2} \)
23 \( 1 + (-0.423 - 0.423i)T + 529iT^{2} \)
29 \( 1 + 11.2iT - 841T^{2} \)
31 \( 1 + 16.4T + 961T^{2} \)
37 \( 1 + (-47.6 + 47.6i)T - 1.36e3iT^{2} \)
41 \( 1 + 44.0T + 1.68e3T^{2} \)
43 \( 1 + (46.7 + 46.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (20.3 - 20.3i)T - 2.20e3iT^{2} \)
53 \( 1 + (18.6 + 18.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 13.4iT - 3.48e3T^{2} \)
61 \( 1 + 10.8T + 3.72e3T^{2} \)
67 \( 1 + (-72.2 + 72.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 64.1T + 5.04e3T^{2} \)
73 \( 1 + (-51.4 - 51.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 157. iT - 6.24e3T^{2} \)
83 \( 1 + (76.9 + 76.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 37.6iT - 7.92e3T^{2} \)
97 \( 1 + (-97.2 + 97.2i)T - 9.40e3iT^{2} \)
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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

−12.55728590769106616420054626416, −11.60442842930074607636636500347, −10.74926373259731792674142711306, −10.00401597173274841501818493159, −8.309412054153323176130357115919, −7.42424347000978726116800062279, −6.18305903677379266597704662416, −5.09677434211087661219120649232, −3.67322233830124865344256363194, −2.35593389668458868843520291710, 0.00275135664856288898731360347, 3.05700146024233257470936507950, 4.54856793771602274289158255176, 5.08969736985828003556101798591, 6.64797810283020436781666522455, 7.57900416457339515589842034166, 8.656247296936516622430856978942, 9.798677622884936738293723520695, 11.08614500692932152642684158837, 11.76467304585616366917659681840

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