Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $-0.281 + 0.959i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (0.866 + 1.5i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−1.5 + 0.866i)6-s + (−2.5 − 0.866i)7-s i·8-s + (−1.5 + 2.59i)9-s + (0.866 − 0.5i)10-s + (−4.09 − 2.36i)11-s + (−0.866 − 1.5i)12-s + (−3 − 1.73i)13-s + (0.866 − 2.5i)14-s + (0.866 − 1.5i)15-s + 16-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.499 + 0.866i)3-s − 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.612 + 0.353i)6-s + (−0.944 − 0.327i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.273 − 0.158i)10-s + (−1.23 − 0.713i)11-s + (−0.249 − 0.433i)12-s + (−0.832 − 0.480i)13-s + (0.231 − 0.668i)14-s + (0.223 − 0.387i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.281 + 0.959i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (131, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ -0.281 + 0.959i)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 - iT \)
3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good11 \( 1 + (4.09 + 2.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.09 + 0.633i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.19 + 1.26i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.59 - 3.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.19iT - 31T^{2} \)
37 \( 1 + (3.09 - 5.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.59 + 2.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9T + 47T^{2} \)
53 \( 1 + (-6.29 + 3.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.19T + 59T^{2} \)
61 \( 1 - 12.9iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 4.73iT - 71T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + (5.59 + 9.69i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.19 + 3.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.39 - 4.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.31579919244191883308378181727, −9.318098855297132481919530786948, −8.618767238160573727136728067158, −7.79517787228108252929192716538, −6.90127023460504338083326593194, −5.51329322145192892682867486972, −4.96785759406228997175136856123, −3.69831776505631669409499329688, −2.82510419271929226058787298081, 0, 2.14615554663578379021417444230, 2.79502513973306058405091944143, 3.96578539162292035218827257301, 5.39415029064606927896043746479, 6.53037265933990513408761694785, 7.45580786820551258123825439594, 8.123947896503976530483251752122, 9.479237533600101899617652561490, 9.705904262587716238925016345933

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