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Properties

Label 2-45-15.8-c3-0-0
Degree $2$
Conductor $45$
Sign $0.939 - 0.343i$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
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  + (−3.74 − 3.74i)2-s + 20.0i·4-s + (−0.572 + 11.1i)5-s + (1.80 − 1.80i)7-s + (45.3 − 45.3i)8-s + (43.9 − 39.7i)10-s + 46.0i·11-s + (18.6 + 18.6i)13-s − 13.5·14-s − 178.·16-s + (14.5 + 14.5i)17-s + 74.4i·19-s + (−224. − 11.4i)20-s + (172. − 172. i)22-s + (−59.6 + 59.6i)23-s + ⋯
L(s)  = 1  + (−1.32 − 1.32i)2-s + 2.51i·4-s + (−0.0511 + 0.998i)5-s + (0.0977 − 0.0977i)7-s + (2.00 − 2.00i)8-s + (1.39 − 1.25i)10-s + 1.26i·11-s + (0.397 + 0.397i)13-s − 0.258·14-s − 2.79·16-s + (0.208 + 0.208i)17-s + 0.899i·19-s + (−2.50 − 0.128i)20-s + (1.67 − 1.67i)22-s + (−0.540 + 0.540i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.939 - 0.343i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.939 - 0.343i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.588678 + 0.104242i\)
\(L(\frac12)\) \(\approx\) \(0.588678 + 0.104242i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.572 - 11.1i)T \)
good2 \( 1 + (3.74 + 3.74i)T + 8iT^{2} \)
7 \( 1 + (-1.80 + 1.80i)T - 343iT^{2} \)
11 \( 1 - 46.0iT - 1.33e3T^{2} \)
13 \( 1 + (-18.6 - 18.6i)T + 2.19e3iT^{2} \)
17 \( 1 + (-14.5 - 14.5i)T + 4.91e3iT^{2} \)
19 \( 1 - 74.4iT - 6.85e3T^{2} \)
23 \( 1 + (59.6 - 59.6i)T - 1.21e4iT^{2} \)
29 \( 1 - 202.T + 2.43e4T^{2} \)
31 \( 1 + 49.5T + 2.97e4T^{2} \)
37 \( 1 + (45.0 - 45.0i)T - 5.06e4iT^{2} \)
41 \( 1 + 306. iT - 6.89e4T^{2} \)
43 \( 1 + (230. + 230. i)T + 7.95e4iT^{2} \)
47 \( 1 + (176. + 176. i)T + 1.03e5iT^{2} \)
53 \( 1 + (85.1 - 85.1i)T - 1.48e5iT^{2} \)
59 \( 1 - 330.T + 2.05e5T^{2} \)
61 \( 1 - 678.T + 2.26e5T^{2} \)
67 \( 1 + (-756. + 756. i)T - 3.00e5iT^{2} \)
71 \( 1 + 100. iT - 3.57e5T^{2} \)
73 \( 1 + (-586. - 586. i)T + 3.89e5iT^{2} \)
79 \( 1 + 286. iT - 4.93e5T^{2} \)
83 \( 1 + (947. - 947. i)T - 5.71e5iT^{2} \)
89 \( 1 - 688.T + 7.04e5T^{2} \)
97 \( 1 + (-920. + 920. i)T - 9.12e5iT^{2} \)
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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

−15.65339151395209392192992529584, −14.08420831094039668511131438445, −12.49557388181094353917183369193, −11.58822489049386009981132234865, −10.42884878875940618777121909860, −9.746269050453220383730326133329, −8.202448894421882339990123829268, −7.00731749598859618935037118104, −3.72445531083315223575738644386, −1.99182188851684270365657268108, 0.75903665111038150249554240269, 5.17356129192272512664797164774, 6.41858366182037909506297088879, 8.119944926655733949459014821605, 8.693878794698791650888906632155, 9.940198332724284220177900782058, 11.35761146131236034241485408161, 13.30221167457494046000289642434, 14.51760242132746772984199682574, 15.91796009652913530905398929551

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