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Properties

Label 2-40-40.29-c5-0-2
Degree $2$
Conductor $40$
Sign $-0.260 - 0.965i$
Analytic cond. $6.41535$
Root an. cond. $2.53285$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.53 + 1.14i)2-s − 16.0·3-s + (29.3 − 12.7i)4-s + (19.1 − 52.5i)5-s + (88.6 − 18.3i)6-s − 20.5i·7-s + (−148. + 104. i)8-s + 13.2·9-s + (−45.6 + 312. i)10-s + 619. i·11-s + (−470. + 203. i)12-s − 101.·13-s + (23.5 + 113. i)14-s + (−306. + 840. i)15-s + (701. − 746. i)16-s + 527. i·17-s + ⋯
L(s)  = 1  + (−0.979 + 0.202i)2-s − 1.02·3-s + (0.917 − 0.397i)4-s + (0.342 − 0.939i)5-s + (1.00 − 0.208i)6-s − 0.158i·7-s + (−0.818 + 0.574i)8-s + 0.0545·9-s + (−0.144 + 0.989i)10-s + 1.54i·11-s + (−0.942 + 0.407i)12-s − 0.166·13-s + (0.0321 + 0.155i)14-s + (−0.351 + 0.964i)15-s + (0.684 − 0.728i)16-s + 0.442i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.260 - 0.965i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ -0.260 - 0.965i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.251343 + 0.328121i\)
\(L(\frac12)\) \(\approx\) \(0.251343 + 0.328121i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.53 - 1.14i)T \)
5 \( 1 + (-19.1 + 52.5i)T \)
good3 \( 1 + 16.0T + 243T^{2} \)
7 \( 1 + 20.5iT - 1.68e4T^{2} \)
11 \( 1 - 619. iT - 1.61e5T^{2} \)
13 \( 1 + 101.T + 3.71e5T^{2} \)
17 \( 1 - 527. iT - 1.41e6T^{2} \)
19 \( 1 - 1.70e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.54e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.30e3iT - 2.05e7T^{2} \)
31 \( 1 - 201.T + 2.86e7T^{2} \)
37 \( 1 - 1.24e4T + 6.93e7T^{2} \)
41 \( 1 + 1.39e4T + 1.15e8T^{2} \)
43 \( 1 + 1.62e4T + 1.47e8T^{2} \)
47 \( 1 - 2.99e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.35e4T + 4.18e8T^{2} \)
59 \( 1 + 1.37e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.03e4iT - 8.44e8T^{2} \)
67 \( 1 + 7.81e3T + 1.35e9T^{2} \)
71 \( 1 - 4.04e3T + 1.80e9T^{2} \)
73 \( 1 + 4.94e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.87e4T + 3.07e9T^{2} \)
83 \( 1 + 7.59e3T + 3.93e9T^{2} \)
89 \( 1 - 8.00e4T + 5.58e9T^{2} \)
97 \( 1 - 1.35e5iT - 8.58e9T^{2} \)
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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

−15.96390153081587984732954640581, −14.63573249359142567681483475521, −12.65916664203900974764929141622, −11.81118910483895718628123304965, −10.40531271651203811954628413008, −9.415214697483831821807667344112, −7.88613085950501819823991910660, −6.33564409985816696116620239274, −5.03229452800335530598496683353, −1.51213836751338167783396529427, 0.36372883770281457497661546948, 2.84424170641780444406017399224, 5.82508979724821203063731876316, 6.84447458463056346856479150998, 8.563160256434827766889305730969, 10.09675724651360977595597976074, 11.17352798098303619504180617251, 11.69059171584402860179164443932, 13.53547031473431033620187543370, 15.11536678764570620466274365634

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