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Properties

Label 2-45-15.2-c21-0-15
Degree $2$
Conductor $45$
Sign $0.0361 - 0.999i$
Analytic cond. $125.764$
Root an. cond. $11.2144$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.63e3 + 1.63e3i)2-s − 3.24e6i·4-s + (5.62e5 + 2.18e7i)5-s + (3.62e8 + 3.62e8i)7-s + (1.86e9 + 1.86e9i)8-s + (−3.65e10 − 3.47e10i)10-s − 1.00e11i·11-s + (3.85e9 − 3.85e9i)13-s − 1.18e12·14-s + 6.93e11·16-s + (−8.99e12 + 8.99e12i)17-s − 3.16e13i·19-s + (7.07e13 − 1.82e12i)20-s + (1.63e14 + 1.63e14i)22-s + (1.75e14 + 1.75e14i)23-s + ⋯
L(s)  = 1  + (−1.12 + 1.12i)2-s − 1.54i·4-s + (0.0257 + 0.999i)5-s + (0.485 + 0.485i)7-s + (0.614 + 0.614i)8-s + (−1.15 − 1.09i)10-s − 1.16i·11-s + (0.00775 − 0.00775i)13-s − 1.09·14-s + 0.157·16-s + (−1.08 + 1.08i)17-s − 1.18i·19-s + (1.54 − 0.0398i)20-s + (1.31 + 1.31i)22-s + (0.883 + 0.883i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0361 - 0.999i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.0361 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.0361 - 0.999i$
Analytic conductor: \(125.764\)
Root analytic conductor: \(11.2144\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :21/2),\ 0.0361 - 0.999i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.9483541297\)
\(L(\frac12)\) \(\approx\) \(0.9483541297\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-5.62e5 - 2.18e7i)T \)
good2 \( 1 + (1.63e3 - 1.63e3i)T - 2.09e6iT^{2} \)
7 \( 1 + (-3.62e8 - 3.62e8i)T + 5.58e17iT^{2} \)
11 \( 1 + 1.00e11iT - 7.40e21T^{2} \)
13 \( 1 + (-3.85e9 + 3.85e9i)T - 2.47e23iT^{2} \)
17 \( 1 + (8.99e12 - 8.99e12i)T - 6.90e25iT^{2} \)
19 \( 1 + 3.16e13iT - 7.14e26T^{2} \)
23 \( 1 + (-1.75e14 - 1.75e14i)T + 3.94e28iT^{2} \)
29 \( 1 - 1.14e15T + 5.13e30T^{2} \)
31 \( 1 + 6.89e15T + 2.08e31T^{2} \)
37 \( 1 + (1.65e16 + 1.65e16i)T + 8.55e32iT^{2} \)
41 \( 1 - 2.61e16iT - 7.38e33T^{2} \)
43 \( 1 + (-4.43e16 + 4.43e16i)T - 2.00e34iT^{2} \)
47 \( 1 + (6.90e16 - 6.90e16i)T - 1.30e35iT^{2} \)
53 \( 1 + (3.02e17 + 3.02e17i)T + 1.62e36iT^{2} \)
59 \( 1 - 4.99e18T + 1.54e37T^{2} \)
61 \( 1 - 8.37e18T + 3.10e37T^{2} \)
67 \( 1 + (-8.33e17 - 8.33e17i)T + 2.22e38iT^{2} \)
71 \( 1 + 3.96e19iT - 7.52e38T^{2} \)
73 \( 1 + (5.48e18 - 5.48e18i)T - 1.34e39iT^{2} \)
79 \( 1 + 1.46e20iT - 7.08e39T^{2} \)
83 \( 1 + (-6.35e19 - 6.35e19i)T + 1.99e40iT^{2} \)
89 \( 1 - 2.65e20T + 8.65e40T^{2} \)
97 \( 1 + (-5.03e20 - 5.03e20i)T + 5.27e41iT^{2} \)
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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

−11.33612473253125522546135082501, −10.63686357608316407574991339245, −9.172727706207426481533572600184, −8.448059978871575319388505504204, −7.26757415111667164995438507205, −6.39496775389322440801793364650, −5.39464867638840580102046001789, −3.43973601039400461863162431534, −1.98450135995784348423092709301, −0.50124979094673976203593152411, 0.59039937448873308842346508170, 1.49879987671959762573457001796, 2.37835848538027332113862321314, 4.01604241391013526453004454815, 5.09103048040951646235304820842, 7.15667256150252824668338338182, 8.304432083609582356935001896951, 9.204611375913485733672228328316, 10.10862580041939630755549178916, 11.19468928117421114462017787042

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