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Properties

Label 2-45-15.8-c3-0-1
Degree $2$
Conductor $45$
Sign $0.314 - 0.949i$
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  + (2.62 + 2.62i)2-s + 5.74i·4-s + (8.38 + 7.38i)5-s + (−10.8 + 10.8i)7-s + (5.91 − 5.91i)8-s + (2.62 + 41.3i)10-s − 37.8i·11-s + (−48.1 − 48.1i)13-s − 56.9·14-s + 76.9·16-s + (60.3 + 60.3i)17-s − 109. i·19-s + (−42.4 + 48.1i)20-s + (99.3 − 99.3i)22-s + (−39.7 + 39.7i)23-s + ⋯
L(s)  = 1  + (0.926 + 0.926i)2-s + 0.717i·4-s + (0.750 + 0.660i)5-s + (−0.586 + 0.586i)7-s + (0.261 − 0.261i)8-s + (0.0829 + 1.30i)10-s − 1.03i·11-s + (−1.02 − 1.02i)13-s − 1.08·14-s + 1.20·16-s + (0.860 + 0.860i)17-s − 1.32i·19-s + (−0.474 + 0.538i)20-s + (0.962 − 0.962i)22-s + (−0.360 + 0.360i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.314 - 0.949i$
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: \((2,\ 45,\ (\ :3/2),\ 0.314 - 0.949i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.72859 + 1.24773i\)
\(L(\frac12)\) \(\approx\) \(1.72859 + 1.24773i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.38 - 7.38i)T \)
good2 \( 1 + (-2.62 - 2.62i)T + 8iT^{2} \)
7 \( 1 + (10.8 - 10.8i)T - 343iT^{2} \)
11 \( 1 + 37.8iT - 1.33e3T^{2} \)
13 \( 1 + (48.1 + 48.1i)T + 2.19e3iT^{2} \)
17 \( 1 + (-60.3 - 60.3i)T + 4.91e3iT^{2} \)
19 \( 1 + 109. iT - 6.85e3T^{2} \)
23 \( 1 + (39.7 - 39.7i)T - 1.21e4iT^{2} \)
29 \( 1 + 90.3T + 2.43e4T^{2} \)
31 \( 1 + 233.T + 2.97e4T^{2} \)
37 \( 1 + (19.0 - 19.0i)T - 5.06e4iT^{2} \)
41 \( 1 - 260. iT - 6.89e4T^{2} \)
43 \( 1 + (-176. - 176. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-145. - 145. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-183. + 183. i)T - 1.48e5iT^{2} \)
59 \( 1 + 279.T + 2.05e5T^{2} \)
61 \( 1 + 390.T + 2.26e5T^{2} \)
67 \( 1 + (-150. + 150. i)T - 3.00e5iT^{2} \)
71 \( 1 + 470. iT - 3.57e5T^{2} \)
73 \( 1 + (-480. - 480. i)T + 3.89e5iT^{2} \)
79 \( 1 + 1.32e3iT - 4.93e5T^{2} \)
83 \( 1 + (456. - 456. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + (785. - 785. i)T - 9.12e5iT^{2} \)
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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

−15.26016071121007105197076069209, −14.58044212505207457202034949645, −13.45278308670609979192329405693, −12.61251293289889370632330716980, −10.76298896797891811995093856464, −9.499034826711465538520732738614, −7.59467013565551106424813063455, −6.19743669177156941806920804516, −5.41418334561362136170311261239, −3.14808973203665338653496862296, 2.02030385586885864426110441074, 4.05164525542995109305079542601, 5.37674474184300651466341219663, 7.34074955931354694463574678388, 9.477576943026715425871656160743, 10.32805542016596991124046485695, 12.09683374053907862845147383542, 12.57308270414617900961702824107, 13.81448100153232719771506628971, 14.50082827412557969783130956615

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