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Properties

Label 2-450-25.9-c1-0-8
Degree $2$
Conductor $450$
Sign $0.345 + 0.938i$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (1.44 − 1.70i)5-s − 1.27i·7-s + (0.951 + 0.309i)8-s + (0.530 + 2.17i)10-s + (−3.52 − 2.56i)11-s + (−2.51 − 3.46i)13-s + (1.03 + 0.748i)14-s + (−0.809 + 0.587i)16-s + (0.0930 + 0.0302i)17-s + (0.103 − 0.317i)19-s + (−2.06 − 0.847i)20-s + (4.14 − 1.34i)22-s + (−3.71 + 5.10i)23-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.154 − 0.475i)4-s + (0.646 − 0.762i)5-s − 0.481i·7-s + (0.336 + 0.109i)8-s + (0.167 + 0.686i)10-s + (−1.06 − 0.773i)11-s + (−0.698 − 0.961i)13-s + (0.275 + 0.200i)14-s + (−0.202 + 0.146i)16-s + (0.0225 + 0.00733i)17-s + (0.0236 − 0.0728i)19-s + (−0.462 − 0.189i)20-s + (0.884 − 0.287i)22-s + (−0.773 + 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.345 + 0.938i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.345 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.787665 - 0.549126i\)
\(L(\frac12)\) \(\approx\) \(0.787665 - 0.549126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 \)
5 \( 1 + (-1.44 + 1.70i)T \)
good7 \( 1 + 1.27iT - 7T^{2} \)
11 \( 1 + (3.52 + 2.56i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.51 + 3.46i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.0930 - 0.0302i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.103 + 0.317i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.71 - 5.10i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.44 + 4.44i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.48 + 7.64i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.41 - 3.32i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.24 + 6.71i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 + (-11.1 + 3.61i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.841 + 0.273i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.15 - 4.47i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.35 - 0.985i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.11 - 0.361i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.728 - 2.24i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.945 - 1.30i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.51 + 7.73i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (7.05 + 2.29i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (1.38 + 1.00i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-6.28 + 2.04i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.58127682725929459748580919825, −9.946859462086480215813321757554, −9.129644827094684617600306011914, −7.971403323716969283046044097108, −7.59465768687868578678764809601, −5.95550189737204096097805453437, −5.52326140774085637234237657318, −4.29289680855928684608547287651, −2.51249355014333875321363756653, −0.66808173566470533336057162955, 2.05373170810165585226668172506, 2.77094965907302117364652784623, 4.38789970232017488202271127581, 5.56112840917514037404609373825, 6.79989835434570338423363877015, 7.58189522773278569895233076691, 8.798858348588849983590225589449, 9.630255077845862186849504283429, 10.37425861055400513292821462872, 11.00922577359791053575781817939

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