Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.982 - 0.184i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.397 + 2.61i)7-s + 0.999i·8-s + (−0.429 − 0.248i)11-s + 2.74i·13-s + (−1.65 − 2.06i)14-s + (−0.5 − 0.866i)16-s + (1.82 − 3.16i)17-s + (3.12 − 1.80i)19-s + 0.496·22-s + (5.56 − 3.21i)23-s + (−1.37 − 2.37i)26-s + (2.46 + 0.963i)28-s − 8.87i·29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.150 + 0.988i)7-s + 0.353i·8-s + (−0.129 − 0.0748i)11-s + 0.761i·13-s + (−0.441 − 0.552i)14-s + (−0.125 − 0.216i)16-s + (0.443 − 0.768i)17-s + (0.716 − 0.413i)19-s + 0.105·22-s + (1.16 − 0.669i)23-s + (−0.269 − 0.466i)26-s + (0.465 + 0.182i)28-s − 1.64i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.982 - 0.184i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.982 - 0.184i)$
$L(1)$  $\approx$  $1.360049636$
$L(\frac12)$  $\approx$  $1.360049636$
$L(\frac{3}{2})$   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 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.397 - 2.61i)T \)
good11 \( 1 + (0.429 + 0.248i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.74iT - 13T^{2} \)
17 \( 1 + (-1.82 + 3.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.12 + 1.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.56 + 3.21i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.87iT - 29T^{2} \)
31 \( 1 + (6.90 + 3.98i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.14 + 1.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.22T + 41T^{2} \)
43 \( 1 - 2.22T + 43T^{2} \)
47 \( 1 + (3.27 + 5.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.72 - 3.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.05 + 5.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.24 + 1.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.08 - 7.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-11.3 - 6.53i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.44 - 7.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.79T + 83T^{2} \)
89 \( 1 + (0.743 + 1.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.05iT - 97T^{2} \)
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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

−8.759639498077339746683151538172, −8.018965076644230549652358597700, −7.23547856531910753682580191936, −6.60231782737094481132872496061, −5.61446721765881743905972164995, −5.17146010414070134649215186325, −4.07331621099628642511605133020, −2.79679761406044260020648196558, −2.09052564161447216400683877744, −0.69417938564432717195553395468, 0.908265581835896549599992354691, 1.71710693608844224817117407775, 3.24458208688145027345837519458, 3.54691378556722106206961926772, 4.83130964326800315018004389239, 5.54470329345271795032254559724, 6.64839315759294007225467461548, 7.43987568154186963167316715845, 7.78842330155814653445651827793, 8.741134355619087363697673485274

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