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Properties

Label 2-90-5.3-c6-0-2
Degree $2$
Conductor $90$
Sign $-0.767 - 0.640i$
Analytic cond. $20.7048$
Root an. cond. $4.55026$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 4i)2-s − 32i·4-s + (75 + 100i)5-s + (−247 + 247i)7-s + (−128 − 128i)8-s + (700 + 100i)10-s − 1.40e3·11-s + (−2.70e3 − 2.70e3i)13-s + 1.97e3i·14-s − 1.02e3·16-s + (−2.59e3 + 2.59e3i)17-s − 1.72e3i·19-s + (3.20e3 − 2.40e3i)20-s + (−5.60e3 + 5.60e3i)22-s + (−2.13e3 − 2.13e3i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.5i·4-s + (0.599 + 0.800i)5-s + (−0.720 + 0.720i)7-s + (−0.250 − 0.250i)8-s + (0.700 + 0.100i)10-s − 1.05·11-s + (−1.23 − 1.23i)13-s + 0.720i·14-s − 0.250·16-s + (−0.527 + 0.527i)17-s − 0.250i·19-s + (0.400 − 0.299i)20-s + (−0.526 + 0.526i)22-s + (−0.175 − 0.175i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.767 - 0.640i$
Analytic conductor: \(20.7048\)
Root analytic conductor: \(4.55026\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :3),\ -0.767 - 0.640i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.146057 + 0.402957i\)
\(L(\frac12)\) \(\approx\) \(0.146057 + 0.402957i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 + 4i)T \)
3 \( 1 \)
5 \( 1 + (-75 - 100i)T \)
good7 \( 1 + (247 - 247i)T - 1.17e5iT^{2} \)
11 \( 1 + 1.40e3T + 1.77e6T^{2} \)
13 \( 1 + (2.70e3 + 2.70e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (2.59e3 - 2.59e3i)T - 2.41e7iT^{2} \)
19 \( 1 + 1.72e3iT - 4.70e7T^{2} \)
23 \( 1 + (2.13e3 + 2.13e3i)T + 1.48e8iT^{2} \)
29 \( 1 - 3.05e4iT - 5.94e8T^{2} \)
31 \( 1 + 3.78e4T + 8.87e8T^{2} \)
37 \( 1 + (-3.71e4 + 3.71e4i)T - 2.56e9iT^{2} \)
41 \( 1 - 3.54e4T + 4.75e9T^{2} \)
43 \( 1 + (-3.91e4 - 3.91e4i)T + 6.32e9iT^{2} \)
47 \( 1 + (9.51e4 - 9.51e4i)T - 1.07e10iT^{2} \)
53 \( 1 + (3.60e4 + 3.60e4i)T + 2.21e10iT^{2} \)
59 \( 1 + 3.59e4iT - 4.21e10T^{2} \)
61 \( 1 - 8.33e4T + 5.15e10T^{2} \)
67 \( 1 + (-6.08e4 + 6.08e4i)T - 9.04e10iT^{2} \)
71 \( 1 - 4.03e4T + 1.28e11T^{2} \)
73 \( 1 + (1.29e5 + 1.29e5i)T + 1.51e11iT^{2} \)
79 \( 1 + 5.24e5iT - 2.43e11T^{2} \)
83 \( 1 + (-1.14e5 - 1.14e5i)T + 3.26e11iT^{2} \)
89 \( 1 - 1.87e5iT - 4.96e11T^{2} \)
97 \( 1 + (-5.32e5 + 5.32e5i)T - 8.32e11iT^{2} \)
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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

−12.90659162306233259521310846468, −12.72080597963365331754577819678, −11.04082163577976276153011677282, −10.26051239797065738713836479426, −9.284889288292758594728136392404, −7.52732887570071554203270932564, −6.11846653768333266733256589490, −5.16219390518300436757912977727, −3.12002662920330230389779545959, −2.31554495773752701089792040816, 0.11476818580777645209833985475, 2.32740584311359197384466324035, 4.21443480897899988040819078917, 5.29347774804405597401155189830, 6.64327330575763261087224963082, 7.73875061616226159097258905986, 9.231953706310146854747662293979, 10.08364927269218266355360909840, 11.70760239852399642527844274708, 12.87212511669701196809198595644

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