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Properties

Label 2-450-225.106-c1-0-0
Degree $2$
Conductor $450$
Sign $0.214 - 0.976i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.978 − 0.207i)2-s + (−1.01 − 1.40i)3-s + (0.913 + 0.406i)4-s + (−2.04 − 0.909i)5-s + (0.704 + 1.58i)6-s + (0.118 − 0.204i)7-s + (−0.809 − 0.587i)8-s + (−0.927 + 2.85i)9-s + (1.80 + 1.31i)10-s + (−1.35 − 0.287i)11-s + (−0.360 − 1.69i)12-s + (−2.32 + 0.495i)13-s + (−0.157 + 0.175i)14-s + (0.805 + 3.78i)15-s + (0.669 + 0.743i)16-s + (−1.42 − 1.03i)17-s + ⋯
L(s)  = 1  + (−0.691 − 0.147i)2-s + (−0.587 − 0.809i)3-s + (0.456 + 0.203i)4-s + (−0.913 − 0.406i)5-s + (0.287 + 0.645i)6-s + (0.0446 − 0.0772i)7-s + (−0.286 − 0.207i)8-s + (−0.309 + 0.951i)9-s + (0.572 + 0.415i)10-s + (−0.407 − 0.0866i)11-s + (−0.103 − 0.489i)12-s + (−0.646 + 0.137i)13-s + (−0.0422 + 0.0468i)14-s + (0.207 + 0.978i)15-s + (0.167 + 0.185i)16-s + (−0.346 − 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.214 - 0.976i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.214 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.191729 + 0.154154i\)
\(L(\frac12)\) \(\approx\) \(0.191729 + 0.154154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.978 + 0.207i)T \)
3 \( 1 + (1.01 + 1.40i)T \)
5 \( 1 + (2.04 + 0.909i)T \)
good7 \( 1 + (-0.118 + 0.204i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.35 + 0.287i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (2.32 - 0.495i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (1.42 + 1.03i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.42 - 2.48i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.49 - 1.66i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (-0.482 - 4.59i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (0.701 - 6.67i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-1.47 + 4.53i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.373 - 0.0794i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (1.73 - 3.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.741 - 7.05i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (-0.0729 + 0.0530i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.461 - 0.0981i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (14.1 + 2.99i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (0.298 - 2.83i)T + (-65.5 - 13.9i)T^{2} \)
71 \( 1 + (7.09 - 5.15i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.85 + 5.70i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.581 - 5.53i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (2.65 - 1.18i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (2.07 + 6.37i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-0.258 - 2.45i)T + (-94.8 + 20.1i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.25873954183964763436026461457, −10.62679041706281480289971251745, −9.416272332769647834013526688987, −8.391535908362537155757105238291, −7.58253655336291583422233621141, −7.04682876608009870847379236572, −5.70862944861449556265209159357, −4.62814794500165703331570835366, −2.99736953270184490175547640649, −1.36546221158314492737192627328, 0.21776491701612889018211343021, 2.74459997177169095393460504703, 4.06914477831822584137186598888, 5.13612808920087622651803008998, 6.30671409265060516958037295319, 7.30158088066101574978852683476, 8.176008366387180743659390271924, 9.217566981125310180750813736906, 10.10770512507003744658371165727, 10.75118120712944699631829783511

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