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Properties

Label 2-45-45.14-c4-0-9
Degree $2$
Conductor $45$
Sign $0.999 - 0.0228i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.02 + 1.76i)2-s + (−8.99 − 0.0407i)3-s + (5.91 − 10.2i)4-s + (24.7 + 3.55i)5-s + (−9.12 − 15.9i)6-s + (2.09 − 1.21i)7-s + 56.8·8-s + (80.9 + 0.732i)9-s + (18.9 + 47.4i)10-s + (117. − 67.9i)11-s + (−53.6 + 91.9i)12-s + (136. + 78.8i)13-s + (4.28 + 2.47i)14-s + (−222. − 32.9i)15-s + (−36.4 − 63.1i)16-s − 258.·17-s + ⋯
L(s)  = 1  + (0.255 + 0.442i)2-s + (−0.999 − 0.00452i)3-s + (0.369 − 0.639i)4-s + (0.989 + 0.142i)5-s + (−0.253 − 0.443i)6-s + (0.0428 − 0.0247i)7-s + 0.888·8-s + (0.999 + 0.00904i)9-s + (0.189 + 0.474i)10-s + (0.972 − 0.561i)11-s + (−0.372 + 0.638i)12-s + (0.807 + 0.466i)13-s + (0.0218 + 0.0126i)14-s + (−0.989 − 0.146i)15-s + (−0.142 − 0.246i)16-s − 0.893·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0228i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.0228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.999 - 0.0228i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.999 - 0.0228i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.70367 + 0.0195041i\)
\(L(\frac12)\) \(\approx\) \(1.70367 + 0.0195041i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.99 + 0.0407i)T \)
5 \( 1 + (-24.7 - 3.55i)T \)
good2 \( 1 + (-1.02 - 1.76i)T + (-8 + 13.8i)T^{2} \)
7 \( 1 + (-2.09 + 1.21i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-117. + 67.9i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-136. - 78.8i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 258.T + 8.35e4T^{2} \)
19 \( 1 + 363.T + 1.30e5T^{2} \)
23 \( 1 + (-392. + 680. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (629. - 363. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (463. - 802. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.99e3iT - 1.87e6T^{2} \)
41 \( 1 + (-1.25e3 - 722. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (542. - 312. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (188. + 326. i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 694.T + 7.89e6T^{2} \)
59 \( 1 + (2.22e3 + 1.28e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (2.57e3 + 4.46e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (5.67e3 + 3.27e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 3.33e3iT - 2.54e7T^{2} \)
73 \( 1 + 1.73e3iT - 2.83e7T^{2} \)
79 \( 1 + (3.69e3 + 6.40e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-5.46e3 - 9.46e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 2.32e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.42e4 - 8.23e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−15.08157294678965612523338506378, −14.01972488505253195127942411289, −12.92055163599657464349457198370, −11.21760267334923928934258595833, −10.58967825251422079397038748763, −9.100638443022840780291280505122, −6.61396557260340331915620398490, −6.24634847359830909825876594455, −4.72575721239076132734701773672, −1.47833748528748518921601505521, 1.77110649550880812910891278033, 4.16256348037995012705605086696, 5.89623624455689239723788824370, 7.14479339263291004663052929292, 9.164188463193841589979520008849, 10.64763772298372008871683079826, 11.49632082220034055949257672516, 12.76055530533466173135795147405, 13.38446000073405600303144262063, 15.18901606861002332145597234480

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