Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $0.959 - 0.282i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.891 + 0.453i)2-s + (−1.69 − 0.333i)3-s + (0.587 + 0.809i)4-s + (1.62 − 1.53i)5-s + (−1.36 − 1.06i)6-s + (2.58 + 2.58i)7-s + (0.156 + 0.987i)8-s + (2.77 + 1.13i)9-s + (2.14 − 0.633i)10-s + (1.45 − 0.473i)11-s + (−0.729 − 1.57i)12-s + (−2.28 − 4.48i)13-s + (1.12 + 3.47i)14-s + (−3.27 + 2.07i)15-s + (−0.309 + 0.951i)16-s + (−5.09 + 0.806i)17-s + ⋯
L(s)  = 1  + (0.630 + 0.321i)2-s + (−0.981 − 0.192i)3-s + (0.293 + 0.404i)4-s + (0.725 − 0.687i)5-s + (−0.556 − 0.436i)6-s + (0.976 + 0.976i)7-s + (0.0553 + 0.349i)8-s + (0.925 + 0.378i)9-s + (0.678 − 0.200i)10-s + (0.439 − 0.142i)11-s + (−0.210 − 0.453i)12-s + (−0.633 − 1.24i)13-s + (0.301 + 0.928i)14-s + (−0.844 + 0.535i)15-s + (−0.0772 + 0.237i)16-s + (−1.23 + 0.195i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.959 - 0.282i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ 0.959 - 0.282i)$
$L(1)$  $\approx$  $1.36023 + 0.196253i$
$L(\frac12)$  $\approx$  $1.36023 + 0.196253i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (-0.891 - 0.453i)T \)
3 \( 1 + (1.69 + 0.333i)T \)
5 \( 1 + (-1.62 + 1.53i)T \)
good7 \( 1 + (-2.58 - 2.58i)T + 7iT^{2} \)
11 \( 1 + (-1.45 + 0.473i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.28 + 4.48i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (5.09 - 0.806i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (1.27 - 1.75i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.88 - 5.66i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-5.64 + 4.10i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (5.95 + 4.32i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (6.84 - 3.48i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-2.85 - 0.929i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-3.24 + 3.24i)T - 43iT^{2} \)
47 \( 1 + (0.446 - 2.81i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (1.02 + 0.161i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (2.10 - 6.48i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.78 - 5.49i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.527 + 3.33i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (2.18 + 3.00i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.531 + 0.270i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (0.782 + 1.07i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.432 + 2.73i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (1.98 + 6.12i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-13.5 - 2.14i)T + (92.2 + 29.9i)T^{2} \)
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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

−12.93083093342037960440618788324, −12.17908783397502579080644970807, −11.40198671193840130071121909519, −10.18456147414496255341574694789, −8.798162500222379568136441871032, −7.67203594027782225306411489197, −6.10787092990314121595657397752, −5.47828789238426623320928865926, −4.52809015085892531742048115183, −1.99938115650086871880136276126, 1.90458393620085550536203445704, 4.18529424870313336145242282259, 4.98590758467702972680347300231, 6.57134579400241466087583084182, 7.03422864685533135225323723646, 9.187603266151546561996804354741, 10.46367414436682323857869165360, 10.92175438333575275390971669610, 11.81812023777220493861144120542, 12.91030423681915822772791940265

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