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Properties

Label 2-90-5.2-c10-0-14
Degree $2$
Conductor $90$
Sign $0.400 + 0.916i$
Analytic cond. $57.1821$
Root an. cond. $7.56188$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−16 − 16i)2-s + 512i·4-s + (1.87e3 + 2.50e3i)5-s + (−8.40e3 − 8.40e3i)7-s + (8.19e3 − 8.19e3i)8-s + (1.00e4 − 7.00e4i)10-s + 1.73e5·11-s + (−2.32e5 + 2.32e5i)13-s + 2.69e5i·14-s − 2.62e5·16-s + (−1.88e6 − 1.88e6i)17-s − 1.10e6i·19-s + (−1.28e6 + 9.60e5i)20-s + (−2.77e6 − 2.77e6i)22-s + (5.22e6 − 5.22e6i)23-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 0.5i·4-s + (0.600 + 0.800i)5-s + (−0.500 − 0.500i)7-s + (0.250 − 0.250i)8-s + (0.100 − 0.700i)10-s + 1.07·11-s + (−0.626 + 0.626i)13-s + 0.500i·14-s − 0.250·16-s + (−1.32 − 1.32i)17-s − 0.444i·19-s + (−0.400 + 0.300i)20-s + (−0.538 − 0.538i)22-s + (0.812 − 0.812i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.400 + 0.916i$
Analytic conductor: \(57.1821\)
Root analytic conductor: \(7.56188\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :5),\ 0.400 + 0.916i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.20838 - 0.790933i\)
\(L(\frac12)\) \(\approx\) \(1.20838 - 0.790933i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (16 + 16i)T \)
3 \( 1 \)
5 \( 1 + (-1.87e3 - 2.50e3i)T \)
good7 \( 1 + (8.40e3 + 8.40e3i)T + 2.82e8iT^{2} \)
11 \( 1 - 1.73e5T + 2.59e10T^{2} \)
13 \( 1 + (2.32e5 - 2.32e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (1.88e6 + 1.88e6i)T + 2.01e12iT^{2} \)
19 \( 1 + 1.10e6iT - 6.13e12T^{2} \)
23 \( 1 + (-5.22e6 + 5.22e6i)T - 4.14e13iT^{2} \)
29 \( 1 - 2.47e7iT - 4.20e14T^{2} \)
31 \( 1 + 1.00e7T + 8.19e14T^{2} \)
37 \( 1 + (-5.63e7 - 5.63e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 1.53e8T + 1.34e16T^{2} \)
43 \( 1 + (-5.93e7 + 5.93e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (1.72e8 + 1.72e8i)T + 5.25e16iT^{2} \)
53 \( 1 + (1.96e8 - 1.96e8i)T - 1.74e17iT^{2} \)
59 \( 1 + 6.94e8iT - 5.11e17T^{2} \)
61 \( 1 - 9.06e8T + 7.13e17T^{2} \)
67 \( 1 + (9.62e8 + 9.62e8i)T + 1.82e18iT^{2} \)
71 \( 1 - 3.12e9T + 3.25e18T^{2} \)
73 \( 1 + (6.36e8 - 6.36e8i)T - 4.29e18iT^{2} \)
79 \( 1 + 1.96e9iT - 9.46e18T^{2} \)
83 \( 1 + (-5.18e9 + 5.18e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 7.77e9iT - 3.11e19T^{2} \)
97 \( 1 + (6.40e8 + 6.40e8i)T + 7.37e19iT^{2} \)
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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

−11.57328054895611207037768344024, −10.82271905800188502615220996772, −9.611899585498310887427886863467, −9.032244636271675686966930942909, −7.08495558030953894121985758589, −6.64083451701452249707127318826, −4.61802680315713853560045091243, −3.15990735681177531408575680899, −2.06164336272855706077561458301, −0.53438257236206558610103594612, 0.946436380490067268693730010612, 2.22851583328661518596239351337, 4.20856199032896179708230982553, 5.66354790357435491297412640257, 6.45346371608806624593173024231, 7.982427481322821424221255448225, 9.117989076421433583621586649147, 9.652603886119210251961421149327, 11.06284334688957871107084630113, 12.46991995141410288214899190978

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