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Properties

Label 2-450-15.2-c3-0-4
Degree $2$
Conductor $450$
Sign $-0.749 - 0.662i$
Analytic cond. $26.5508$
Root an. cond. $5.15275$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + 1.41i)2-s − 4.00i·4-s + (23.9 + 23.9i)7-s + (5.65 + 5.65i)8-s − 0.123i·11-s + (−58.9 + 58.9i)13-s − 67.6·14-s − 16.0·16-s + (59.2 − 59.2i)17-s + 12.1i·19-s + (0.174 + 0.174i)22-s + (19.7 + 19.7i)23-s − 166. i·26-s + (95.6 − 95.6i)28-s + 139.·29-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (1.29 + 1.29i)7-s + (0.250 + 0.250i)8-s − 0.00337i·11-s + (−1.25 + 1.25i)13-s − 1.29·14-s − 0.250·16-s + (0.845 − 0.845i)17-s + 0.146i·19-s + (0.00168 + 0.00168i)22-s + (0.179 + 0.179i)23-s − 1.25i·26-s + (0.645 − 0.645i)28-s + 0.892·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.749 - 0.662i$
Analytic conductor: \(26.5508\)
Root analytic conductor: \(5.15275\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ -0.749 - 0.662i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.300534609\)
\(L(\frac12)\) \(\approx\) \(1.300534609\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 1.41i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-23.9 - 23.9i)T + 343iT^{2} \)
11 \( 1 + 0.123iT - 1.33e3T^{2} \)
13 \( 1 + (58.9 - 58.9i)T - 2.19e3iT^{2} \)
17 \( 1 + (-59.2 + 59.2i)T - 4.91e3iT^{2} \)
19 \( 1 - 12.1iT - 6.85e3T^{2} \)
23 \( 1 + (-19.7 - 19.7i)T + 1.21e4iT^{2} \)
29 \( 1 - 139.T + 2.43e4T^{2} \)
31 \( 1 + 71.8T + 2.97e4T^{2} \)
37 \( 1 + (-117. - 117. i)T + 5.06e4iT^{2} \)
41 \( 1 + 196. iT - 6.89e4T^{2} \)
43 \( 1 + (195. - 195. i)T - 7.95e4iT^{2} \)
47 \( 1 + (397. - 397. i)T - 1.03e5iT^{2} \)
53 \( 1 + (129. + 129. i)T + 1.48e5iT^{2} \)
59 \( 1 + 716.T + 2.05e5T^{2} \)
61 \( 1 - 114.T + 2.26e5T^{2} \)
67 \( 1 + (-375. - 375. i)T + 3.00e5iT^{2} \)
71 \( 1 + 232. iT - 3.57e5T^{2} \)
73 \( 1 + (-37.7 + 37.7i)T - 3.89e5iT^{2} \)
79 \( 1 - 57.2iT - 4.93e5T^{2} \)
83 \( 1 + (-783. - 783. i)T + 5.71e5iT^{2} \)
89 \( 1 + 778.T + 7.04e5T^{2} \)
97 \( 1 + (887. + 887. i)T + 9.12e5iT^{2} \)
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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

−11.14064492486395310445147318379, −9.774218048822736061933880934860, −9.218130471917319409258088503588, −8.242547130709321555944849390597, −7.53210323537142931581343553856, −6.43706170700626663483553136830, −5.24120465297710706132420274016, −4.69239409178226704946157884644, −2.63339072244708717249711390384, −1.52129052387018903490508929583, 0.51069473888758438150225140962, 1.67852090606481789877843825530, 3.16931266524067983393726947328, 4.39939054620887448879934031894, 5.31182019781820462709729442195, 6.95158266033601377519972183254, 7.86690492549505074253886472255, 8.235505091431376167092219313986, 9.727883429336639686351074675857, 10.41052098471987558411757849471

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