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Properties

Label 2-450-45.14-c2-0-29
Degree $2$
Conductor $450$
Sign $0.438 + 0.898i$
Analytic cond. $12.2616$
Root an. cond. $3.50165$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)2-s + (−1.73 + 2.44i)3-s + (−0.999 + 1.73i)4-s + (−4.22 − 0.389i)6-s + (−5.49 + 3.17i)7-s − 2.82·8-s + (−2.99 − 8.48i)9-s + (8.17 − 4.71i)11-s + (−2.51 − 5.44i)12-s + (−17.0 − 9.84i)13-s + (−7.77 − 4.48i)14-s + (−2.00 − 3.46i)16-s + 1.90·17-s + (8.27 − 9.67i)18-s − 4.69·19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.577 + 0.816i)3-s + (−0.249 + 0.433i)4-s + (−0.704 − 0.0648i)6-s + (−0.785 + 0.453i)7-s − 0.353·8-s + (−0.333 − 0.942i)9-s + (0.743 − 0.429i)11-s + (−0.209 − 0.454i)12-s + (−1.31 − 0.757i)13-s + (−0.555 − 0.320i)14-s + (−0.125 − 0.216i)16-s + 0.112·17-s + (0.459 − 0.537i)18-s − 0.247·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.438 + 0.898i$
Analytic conductor: \(12.2616\)
Root analytic conductor: \(3.50165\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1),\ 0.438 + 0.898i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3133055904\)
\(L(\frac12)\) \(\approx\) \(0.3133055904\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 + (1.73 - 2.44i)T \)
5 \( 1 \)
good7 \( 1 + (5.49 - 3.17i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-8.17 + 4.71i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (17.0 + 9.84i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 1.90T + 289T^{2} \)
19 \( 1 + 4.69T + 361T^{2} \)
23 \( 1 + (4.71 - 8.17i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-2.84 + 1.64i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-20.5 + 35.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 17.3iT - 1.36e3T^{2} \)
41 \( 1 + (53.5 + 30.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-0.826 + 0.477i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-7.05 - 12.2i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 9.53T + 2.80e3T^{2} \)
59 \( 1 + (79.2 + 45.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-37.5 - 65.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-26.8 - 15.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 85.9iT - 5.04e3T^{2} \)
73 \( 1 + 96.0iT - 5.32e3T^{2} \)
79 \( 1 + (-14.8 - 25.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (43.9 + 76.1i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 41.3iT - 7.92e3T^{2} \)
97 \( 1 + (83.0 - 47.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.62663233690287841905862196642, −9.694187685684945884981140212966, −9.115728563295667301933618903480, −7.909782975888110843232321192667, −6.68989044610692531027169293088, −5.92620135998038415269547218456, −5.08971745016004371485444374116, −3.96931963037255981826496132611, −2.90885602185162006087620277845, −0.12480811659970975557644551784, 1.48096345091695642199033970694, 2.77310529942830853464972971226, 4.24728871458782374114736838911, 5.21245340186159906071138128390, 6.64253978116058821948996953139, 6.88152471242966231991948877769, 8.287402237898519081126771338930, 9.575416942458283381492685151652, 10.19479036942246050922619335623, 11.24726953593797671503107722924

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