Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.841 + 0.539i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.189 − 2.63i)7-s + 0.999·8-s + (−4.67 + 2.69i)11-s + 2.51·13-s + (2.19 + 1.48i)14-s + (−0.5 + 0.866i)16-s + (−3.89 + 2.24i)17-s + (2.48 + 1.43i)19-s − 5.39i·22-s + (0.133 − 0.232i)23-s + (−1.25 + 2.18i)26-s + (−2.38 + 1.15i)28-s − 8.89i·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.0716 − 0.997i)7-s + 0.353·8-s + (−1.40 + 0.813i)11-s + 0.698·13-s + (0.585 + 0.396i)14-s + (−0.125 + 0.216i)16-s + (−0.945 + 0.545i)17-s + (0.568 + 0.328i)19-s − 1.15i·22-s + (0.0279 − 0.0483i)23-s + (−0.246 + 0.427i)26-s + (−0.449 + 0.218i)28-s − 1.65i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.841 + 0.539i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.841 + 0.539i)$
$L(1)$  $\approx$  $1.090562194$
$L(\frac12)$  $\approx$  $1.090562194$
$L(\frac{3}{2})$   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 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.189 + 2.63i)T \)
good11 \( 1 + (4.67 - 2.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.51T + 13T^{2} \)
17 \( 1 + (3.89 - 2.24i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.48 - 1.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.133 + 0.232i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.89iT - 29T^{2} \)
31 \( 1 + (-4.18 + 2.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.64 - 3.25i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.760T + 41T^{2} \)
43 \( 1 - 5.86iT - 43T^{2} \)
47 \( 1 + (-6.92 - 3.99i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.19 - 7.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.33 + 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.27 + 1.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.50 + 4.91i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.76iT - 71T^{2} \)
73 \( 1 + (5.82 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.29 + 7.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.45iT - 83T^{2} \)
89 \( 1 + (-3.98 + 6.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.16T + 97T^{2} \)
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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

−8.274538579758008570568380669995, −7.892789491955467228709619912905, −7.28959225047824141109733816661, −6.37091338016175326466562971291, −5.78956140689443995510835610139, −4.59504769044321458815894646536, −4.28581774186947763733110420463, −2.93572091501154669186683583080, −1.78345879298537019443192179723, −0.47233108661076118682951976091, 0.942275134639527143068762779639, 2.37291064340851149198452852709, 2.84087375161086884281817547333, 3.83978883285480536957565658602, 5.08697394328441418102281236245, 5.46539260416498292468460112212, 6.51968923870480101944219735076, 7.42101177845893519958349338706, 8.288009494146224657173165838310, 8.790404975035082932512725913686

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