Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.0466 + 0.998i$
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 + (−2.62 + 0.295i)7-s + 0.999·8-s + (0.570 − 0.329i)11-s − 6.13·13-s + (1.05 − 2.42i)14-s + (−0.5 + 0.866i)16-s + (4.22 − 2.43i)17-s + (6.30 + 3.63i)19-s + 0.659i·22-s + (−2.29 + 3.98i)23-s + (3.06 − 5.31i)26-s + (1.57 + 2.12i)28-s + 8.09i·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.993 + 0.111i)7-s + 0.353·8-s + (0.172 − 0.0993i)11-s − 1.70·13-s + (0.282 − 0.648i)14-s + (−0.125 + 0.216i)16-s + (1.02 − 0.591i)17-s + (1.44 + 0.834i)19-s + 0.140i·22-s + (−0.479 + 0.830i)23-s + (0.601 − 1.04i)26-s + (0.296 + 0.402i)28-s + 1.50i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0466 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0466 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.0466 + 0.998i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3150,\ (\ :1/2),\ -0.0466 + 0.998i)\)
\(L(1)\)  \(\approx\)  \(0.3622185594\)
\(L(\frac12)\)  \(\approx\)  \(0.3622185594\)
\(L(\frac{3}{2})\)   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 \( 1 + (2.62 - 0.295i)T \)
good11 \( 1 + (-0.570 + 0.329i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.13T + 13T^{2} \)
17 \( 1 + (-4.22 + 2.43i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.30 - 3.63i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.29 - 3.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.09iT - 29T^{2} \)
31 \( 1 + (-0.759 + 0.438i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.75 - 5.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.25T + 41T^{2} \)
43 \( 1 + 9.03iT - 43T^{2} \)
47 \( 1 + (10.3 + 6.00i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.10 + 10.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.06 + 7.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0618 + 0.0357i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.15 - 0.666i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.60iT - 71T^{2} \)
73 \( 1 + (1.41 + 2.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.88 + 5.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.44iT - 83T^{2} \)
89 \( 1 + (-2.66 + 4.61i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−8.379435929674743657532492760997, −7.59800741951768230416159552027, −7.12122224593894344146402147026, −6.35625828795397708352166854804, −5.35404580374935106320545954901, −5.04322626935638884988637797488, −3.58871722930919372670661299726, −2.98110469285108607038726366066, −1.57319044315833144468764199529, −0.14394431067081871138619431870, 1.07768533135982002873353184882, 2.54689718825084644419568667481, 3.00230758467862847846533477038, 4.10780451423670122791360446408, 4.86886469291229976594390613936, 5.89995561598351588069018312788, 6.69069905847246867977701878425, 7.65433942018838990393465367803, 7.937250148892803444261459680063, 9.309124830435127041422303331359

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