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Properties

Label 2-450-45.23-c1-0-0
Degree $2$
Conductor $450$
Sign $-0.386 + 0.922i$
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  + (−0.258 + 0.965i)2-s + (−1.10 + 1.33i)3-s + (−0.866 − 0.499i)4-s + (−1 − 1.41i)6-s + (1.05 + 0.283i)7-s + (0.707 − 0.707i)8-s + (−0.548 − 2.94i)9-s + (−5.44 + 3.14i)11-s + (1.62 − 0.599i)12-s + (−3.34 + 0.896i)13-s + (−0.548 + 0.949i)14-s + (0.500 + 0.866i)16-s + (−3.14 − 3.14i)17-s + (2.99 + 0.233i)18-s − 1.55i·19-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.639 + 0.769i)3-s + (−0.433 − 0.249i)4-s + (−0.408 − 0.577i)6-s + (0.400 + 0.107i)7-s + (0.249 − 0.249i)8-s + (−0.182 − 0.983i)9-s + (−1.64 + 0.948i)11-s + (0.469 − 0.173i)12-s + (−0.928 + 0.248i)13-s + (−0.146 + 0.253i)14-s + (0.125 + 0.216i)16-s + (−0.763 − 0.763i)17-s + (0.704 + 0.0551i)18-s − 0.355i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0665861 - 0.100132i\)
\(L(\frac12)\) \(\approx\) \(0.0665861 - 0.100132i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (1.10 - 1.33i)T \)
5 \( 1 \)
good7 \( 1 + (-1.05 - 0.283i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (5.44 - 3.14i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.34 - 0.896i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (3.14 + 3.14i)T + 17iT^{2} \)
19 \( 1 + 1.55iT - 19T^{2} \)
23 \( 1 + (0.258 + 0.965i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.57 + 2.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.22 + 3.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + (3.39 + 1.96i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.896 - 3.34i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.32 - 8.69i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.61 - 6.61i)T - 53iT^{2} \)
59 \( 1 + (5.90 - 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.72 - 4.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.978 + 3.65i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.635iT - 71T^{2} \)
73 \( 1 + (2.89 + 2.89i)T + 73iT^{2} \)
79 \( 1 + (-2.12 + 1.22i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.531 - 0.142i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.36T + 89T^{2} \)
97 \( 1 + (10.7 + 2.89i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.54031640426914281258507593837, −10.66080352428822974002822598178, −9.837695729496573362611273838379, −9.193031653202815273584551509673, −7.913659537349525309086355529427, −7.16024040042047092298000244499, −5.98943205265320584430355053872, −4.88488864222204943626558000922, −4.56236686244949686314978786577, −2.56843301315613048337724061577, 0.081136362537314018819414080411, 1.85303847517883318616871046497, 3.05653247503336355870404156587, 4.80841345371285740254654329659, 5.53444729481567481286011805442, 6.79894304845014587240176871031, 7.998116855101950377585400060779, 8.335263508098981341308657628177, 9.914668855489797060585562556325, 10.72668070766291067032426529521

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