Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.729 - 0.683i$
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 + (−4.16 − 2.76i)5-s + 2.44·6-s + (−1.87 − 1.87i)7-s + (2 − 2i)8-s − 2.99i·9-s + (1.39 + 6.93i)10-s + 17.6·11-s + (−2.44 − 2.44i)12-s + (−12.3 + 12.3i)13-s + 3.74i·14-s + (8.48 − 1.71i)15-s − 4·16-s + (18.7 + 18.7i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (−0.832 − 0.553i)5-s + 0.408·6-s + (−0.267 − 0.267i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.139 + 0.693i)10-s + 1.60·11-s + (−0.204 − 0.204i)12-s + (−0.953 + 0.953i)13-s + 0.267i·14-s + (0.565 − 0.114i)15-s − 0.250·16-s + (1.10 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.729 - 0.683i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.729 - 0.683i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.708763 + 0.280063i\)
\(L(\frac12)\)  \(\approx\)  \(0.708763 + 0.280063i\)
\(L(2)\)   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 \( 1 + (4.16 + 2.76i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good11 \( 1 - 17.6T + 121T^{2} \)
13 \( 1 + (12.3 - 12.3i)T - 169iT^{2} \)
17 \( 1 + (-18.7 - 18.7i)T + 289iT^{2} \)
19 \( 1 - 25.5iT - 361T^{2} \)
23 \( 1 + (5.90 - 5.90i)T - 529iT^{2} \)
29 \( 1 + 15.3iT - 841T^{2} \)
31 \( 1 - 11.3T + 961T^{2} \)
37 \( 1 + (-25.5 - 25.5i)T + 1.36e3iT^{2} \)
41 \( 1 - 58.8T + 1.68e3T^{2} \)
43 \( 1 + (-0.282 + 0.282i)T - 1.84e3iT^{2} \)
47 \( 1 + (48.6 + 48.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (-3.93 + 3.93i)T - 2.80e3iT^{2} \)
59 \( 1 - 86.2iT - 3.48e3T^{2} \)
61 \( 1 - 29.2T + 3.72e3T^{2} \)
67 \( 1 + (65.7 + 65.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 1.19T + 5.04e3T^{2} \)
73 \( 1 + (87.0 - 87.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 55.0iT - 6.24e3T^{2} \)
83 \( 1 + (91.8 - 91.8i)T - 6.88e3iT^{2} \)
89 \( 1 + 103. iT - 7.92e3T^{2} \)
97 \( 1 + (-87.4 - 87.4i)T + 9.40e3iT^{2} \)
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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

−11.94622429212325343105494379978, −11.55733417642943913677931304381, −10.15740585254561927242564295438, −9.508156162748252131720181162485, −8.437677552894780164534436890017, −7.36450020396851495597017619670, −6.08119539071312548819905826570, −4.34617969650831578672590800157, −3.69218816387769694716025608043, −1.31220516719191800821502814560, 0.61852412234657013608966523889, 2.94037550267987467208196855000, 4.69245340970319970037609074049, 6.07764550631133461961401559953, 7.08275081346513635259276216922, 7.69656795572317364578239455273, 9.028770980463494872248190739792, 9.959994412231403794043385115208, 11.19708654102160028428538395112, 11.87325796381324009335803837798

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