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Properties

Label 2-45-5.4-c7-0-6
Degree $2$
Conductor $45$
Sign $-0.268 - 0.963i$
Analytic cond. $14.0573$
Root an. cond. $3.74931$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.7i·2-s + 12.0·4-s + (−75 − 269. i)5-s + 420. i·7-s + 1.50e3i·8-s + (2.90e3 − 807. i)10-s + 6.82e3·11-s + 1.01e4i·13-s − 4.52e3·14-s − 1.47e4·16-s + 1.56e4i·17-s + 6.86e3·19-s + (−900. − 3.23e3i)20-s + 7.35e4i·22-s + 2.92e4i·23-s + ⋯
L(s)  = 1  + 0.951i·2-s + 0.0937·4-s + (−0.268 − 0.963i)5-s + 0.462i·7-s + 1.04i·8-s + (0.917 − 0.255i)10-s + 1.54·11-s + 1.28i·13-s − 0.440·14-s − 0.897·16-s + 0.774i·17-s + 0.229·19-s + (−0.0251 − 0.0903i)20-s + 1.47i·22-s + 0.500i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.268 - 0.963i$
Analytic conductor: \(14.0573\)
Root analytic conductor: \(3.74931\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :7/2),\ -0.268 - 0.963i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.19504 + 1.57340i\)
\(L(\frac12)\) \(\approx\) \(1.19504 + 1.57340i\)
\(L(\frac{9}{2})\) 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 \( 1 + (75 + 269. i)T \)
good2 \( 1 - 10.7iT - 128T^{2} \)
7 \( 1 - 420. iT - 8.23e5T^{2} \)
11 \( 1 - 6.82e3T + 1.94e7T^{2} \)
13 \( 1 - 1.01e4iT - 6.27e7T^{2} \)
17 \( 1 - 1.56e4iT - 4.10e8T^{2} \)
19 \( 1 - 6.86e3T + 8.93e8T^{2} \)
23 \( 1 - 2.92e4iT - 3.40e9T^{2} \)
29 \( 1 + 2.55e4T + 1.72e10T^{2} \)
31 \( 1 - 8.21e4T + 2.75e10T^{2} \)
37 \( 1 - 2.23e5iT - 9.49e10T^{2} \)
41 \( 1 - 5.33e5T + 1.94e11T^{2} \)
43 \( 1 + 7.08e5iT - 2.71e11T^{2} \)
47 \( 1 - 5.82e3iT - 5.06e11T^{2} \)
53 \( 1 + 5.89e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.43e6T + 2.48e12T^{2} \)
61 \( 1 - 1.38e6T + 3.14e12T^{2} \)
67 \( 1 - 2.71e6iT - 6.06e12T^{2} \)
71 \( 1 - 4.81e5T + 9.09e12T^{2} \)
73 \( 1 + 1.48e6iT - 1.10e13T^{2} \)
79 \( 1 + 1.05e6T + 1.92e13T^{2} \)
83 \( 1 + 2.60e6iT - 2.71e13T^{2} \)
89 \( 1 + 5.64e6T + 4.42e13T^{2} \)
97 \( 1 + 1.20e7iT - 8.07e13T^{2} \)
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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

−14.85850873474031755014631200360, −13.82952943561980649358367504562, −12.17186094012916599059941757749, −11.46333279955598812749814921761, −9.300680191322618217343810914870, −8.444145765490909196772012947001, −6.96331105988894652200716162107, −5.78175651523406429601359754259, −4.22058888768238047874571043759, −1.63558976467562824483261978922, 0.886640889462396556702910244681, 2.75577376786507715416427618552, 3.93174864754339985088084646245, 6.38462632311541072437273801660, 7.51820673238919932581996536550, 9.526725224418246340888049937428, 10.63010394160488768596802700449, 11.45677304368513362163578498724, 12.49156092167258249498568124097, 13.93749531021024401523168836935

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