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Properties

Label 2-40-5.4-c3-0-3
Degree $2$
Conductor $40$
Sign $-0.259 + 0.965i$
Analytic cond. $2.36007$
Root an. cond. $1.53625$
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  − 6.89i·3-s + (−10.7 − 2.89i)5-s − 12.6i·7-s − 20.5·9-s + 59.1·11-s − 42.2i·13-s + (−20 + 74.4i)15-s + 126. i·17-s + 19.1·19-s − 87.5·21-s − 78.3i·23-s + (108. + 62.6i)25-s − 44.1i·27-s + 148.·29-s − 139.·31-s + ⋯
L(s)  = 1  − 1.32i·3-s + (−0.965 − 0.259i)5-s − 0.685i·7-s − 0.762·9-s + 1.62·11-s − 0.900i·13-s + (−0.344 + 1.28i)15-s + 1.80i·17-s + 0.231·19-s − 0.910·21-s − 0.709i·23-s + (0.865 + 0.500i)25-s − 0.314i·27-s + 0.950·29-s − 0.806·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.259 + 0.965i$
Analytic conductor: \(2.36007\)
Root analytic conductor: \(1.53625\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :3/2),\ -0.259 + 0.965i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.689485 - 0.899011i\)
\(L(\frac12)\) \(\approx\) \(0.689485 - 0.899011i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (10.7 + 2.89i)T \)
good3 \( 1 + 6.89iT - 27T^{2} \)
7 \( 1 + 12.6iT - 343T^{2} \)
11 \( 1 - 59.1T + 1.33e3T^{2} \)
13 \( 1 + 42.2iT - 2.19e3T^{2} \)
17 \( 1 - 126. iT - 4.91e3T^{2} \)
19 \( 1 - 19.1T + 6.85e3T^{2} \)
23 \( 1 + 78.3iT - 1.21e4T^{2} \)
29 \( 1 - 148.T + 2.43e4T^{2} \)
31 \( 1 + 139.T + 2.97e4T^{2} \)
37 \( 1 + 66.5iT - 5.06e4T^{2} \)
41 \( 1 + 203.T + 6.89e4T^{2} \)
43 \( 1 + 288. iT - 7.95e4T^{2} \)
47 \( 1 - 360. iT - 1.03e5T^{2} \)
53 \( 1 - 686. iT - 1.48e5T^{2} \)
59 \( 1 - 83.1T + 2.05e5T^{2} \)
61 \( 1 + 208.T + 2.26e5T^{2} \)
67 \( 1 - 192. iT - 3.00e5T^{2} \)
71 \( 1 - 500.T + 3.57e5T^{2} \)
73 \( 1 + 122. iT - 3.89e5T^{2} \)
79 \( 1 - 289.T + 4.93e5T^{2} \)
83 \( 1 - 573. iT - 5.71e5T^{2} \)
89 \( 1 - 565.T + 7.04e5T^{2} \)
97 \( 1 - 643. iT - 9.12e5T^{2} \)
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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

−15.19126786529019228477464338302, −14.03466050130665747156475556107, −12.71976905285300376393999388642, −12.09991878963809816989025226053, −10.68580998390805873617843002480, −8.606053448484959275163106501689, −7.53219239950439294717541199492, −6.39672957087295814508491037584, −3.93459838131123858317257446246, −1.10271753024097661636281778025, 3.54492742207040441089086225937, 4.82266239270148470286545803042, 6.91109308591638454018750009951, 8.862201353506884773633177360891, 9.685639651378208100840735886915, 11.40042678631306457050497141872, 11.88210507024598909913707012283, 14.11791996426496532407792920388, 15.02728906019259127282902104567, 15.97765666141108400510730768198

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