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Properties

Label 2-45-15.2-c21-0-14
Degree $2$
Conductor $45$
Sign $-0.726 - 0.687i$
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  + (−88.2 + 88.2i)2-s + 2.08e6i·4-s + (1.67e7 + 1.40e7i)5-s + (6.37e8 + 6.37e8i)7-s + (−3.68e8 − 3.68e8i)8-s + (−2.71e9 + 2.42e8i)10-s − 1.46e11i·11-s + (−1.71e11 + 1.71e11i)13-s − 1.12e11·14-s − 4.30e12·16-s + (5.82e12 − 5.82e12i)17-s + 2.23e13i·19-s + (−2.91e13 + 3.48e13i)20-s + (1.29e13 + 1.29e13i)22-s + (2.49e14 + 2.49e14i)23-s + ⋯
L(s)  = 1  + (−0.0609 + 0.0609i)2-s + 0.992i·4-s + (0.767 + 0.641i)5-s + (0.853 + 0.853i)7-s + (−0.121 − 0.121i)8-s + (−0.0858 + 0.00767i)10-s − 1.70i·11-s + (−0.344 + 0.344i)13-s − 0.104·14-s − 0.977·16-s + (0.700 − 0.700i)17-s + 0.835i·19-s + (−0.636 + 0.761i)20-s + (0.104 + 0.104i)22-s + (1.25 + 1.25i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.726 - 0.687i$
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: \((2,\ 45,\ (\ :21/2),\ -0.726 - 0.687i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.653793787\)
\(L(\frac12)\) \(\approx\) \(2.653793787\)
\(L(\frac{23}{2})\) 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 \( 1 + (-1.67e7 - 1.40e7i)T \)
good2 \( 1 + (88.2 - 88.2i)T - 2.09e6iT^{2} \)
7 \( 1 + (-6.37e8 - 6.37e8i)T + 5.58e17iT^{2} \)
11 \( 1 + 1.46e11iT - 7.40e21T^{2} \)
13 \( 1 + (1.71e11 - 1.71e11i)T - 2.47e23iT^{2} \)
17 \( 1 + (-5.82e12 + 5.82e12i)T - 6.90e25iT^{2} \)
19 \( 1 - 2.23e13iT - 7.14e26T^{2} \)
23 \( 1 + (-2.49e14 - 2.49e14i)T + 3.94e28iT^{2} \)
29 \( 1 + 2.46e15T + 5.13e30T^{2} \)
31 \( 1 - 6.47e15T + 2.08e31T^{2} \)
37 \( 1 + (-3.36e16 - 3.36e16i)T + 8.55e32iT^{2} \)
41 \( 1 + 6.87e16iT - 7.38e33T^{2} \)
43 \( 1 + (1.55e17 - 1.55e17i)T - 2.00e34iT^{2} \)
47 \( 1 + (-5.27e16 + 5.27e16i)T - 1.30e35iT^{2} \)
53 \( 1 + (-1.53e17 - 1.53e17i)T + 1.62e36iT^{2} \)
59 \( 1 + 5.42e18T + 1.54e37T^{2} \)
61 \( 1 + 6.13e18T + 3.10e37T^{2} \)
67 \( 1 + (-1.80e19 - 1.80e19i)T + 2.22e38iT^{2} \)
71 \( 1 - 2.47e19iT - 7.52e38T^{2} \)
73 \( 1 + (8.19e18 - 8.19e18i)T - 1.34e39iT^{2} \)
79 \( 1 + 1.12e19iT - 7.08e39T^{2} \)
83 \( 1 + (-6.21e19 - 6.21e19i)T + 1.99e40iT^{2} \)
89 \( 1 - 3.66e20T + 8.65e40T^{2} \)
97 \( 1 + (-6.32e19 - 6.32e19i)T + 5.27e41iT^{2} \)
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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

−11.77244038461193120915011814768, −11.22778473994480772234657062206, −9.576822320623108374386801983008, −8.531923126593003161500719055799, −7.55239953453189538733386852745, −6.16181506204129394171712976218, −5.13717564268146393560930440441, −3.37007886265582003451406192536, −2.65063792302329717677692594403, −1.29513731123489126125512448377, 0.55720603223889480793093386240, 1.41874210645177708049879432038, 2.29121245689482727838676087994, 4.56905861558606992971300001194, 4.98864247595751286493305270954, 6.42524994482635073741317675640, 7.65190132775092076913695844966, 9.159089525141175216180058022965, 10.07561165772763710803916464148, 10.85073134526262861154457055011

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