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Properties

Label 2-40-40.29-c5-0-20
Degree $2$
Conductor $40$
Sign $-0.520 + 0.853i$
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  + (−5.08 + 2.47i)2-s + 10.5·3-s + (19.7 − 25.1i)4-s + (−52.7 − 18.4i)5-s + (−53.6 + 26.1i)6-s + 47.9i·7-s + (−37.9 + 176. i)8-s − 131.·9-s + (314. − 36.8i)10-s − 690. i·11-s + (208. − 265. i)12-s − 743.·13-s + (−118. − 244. i)14-s + (−557. − 194. i)15-s + (−245. − 994. i)16-s − 2.04e3i·17-s + ⋯
L(s)  = 1  + (−0.899 + 0.437i)2-s + 0.677·3-s + (0.616 − 0.787i)4-s + (−0.943 − 0.330i)5-s + (−0.608 + 0.296i)6-s + 0.370i·7-s + (−0.209 + 0.977i)8-s − 0.541·9-s + (0.993 − 0.116i)10-s − 1.72i·11-s + (0.417 − 0.533i)12-s − 1.21·13-s + (−0.162 − 0.332i)14-s + (−0.639 − 0.223i)15-s + (−0.239 − 0.970i)16-s − 1.71i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.520 + 0.853i$
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: \((2,\ 40,\ (\ :5/2),\ -0.520 + 0.853i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.219865 - 0.391599i\)
\(L(\frac12)\) \(\approx\) \(0.219865 - 0.391599i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.08 - 2.47i)T \)
5 \( 1 + (52.7 + 18.4i)T \)
good3 \( 1 - 10.5T + 243T^{2} \)
7 \( 1 - 47.9iT - 1.68e4T^{2} \)
11 \( 1 + 690. iT - 1.61e5T^{2} \)
13 \( 1 + 743.T + 3.71e5T^{2} \)
17 \( 1 + 2.04e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.44e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.43e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.38e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.82e3T + 2.86e7T^{2} \)
37 \( 1 + 632.T + 6.93e7T^{2} \)
41 \( 1 - 9.13e3T + 1.15e8T^{2} \)
43 \( 1 + 1.99e3T + 1.47e8T^{2} \)
47 \( 1 - 1.29e3iT - 2.29e8T^{2} \)
53 \( 1 - 7.76e3T + 4.18e8T^{2} \)
59 \( 1 + 7.01e3iT - 7.14e8T^{2} \)
61 \( 1 - 5.00e3iT - 8.44e8T^{2} \)
67 \( 1 - 1.45e4T + 1.35e9T^{2} \)
71 \( 1 + 5.97e4T + 1.80e9T^{2} \)
73 \( 1 + 3.77e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.81e4T + 3.07e9T^{2} \)
83 \( 1 + 8.70e3T + 3.93e9T^{2} \)
89 \( 1 + 7.88e4T + 5.58e9T^{2} \)
97 \( 1 - 5.86e3iT - 8.58e9T^{2} \)
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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

−14.84184835939298307404553896982, −14.04778616569560277138376105753, −12.04214727488827022202383631490, −11.03699230637840080345689261646, −9.256575373632826000702516690725, −8.448689624315964423156357508029, −7.39093647843997858539871217961, −5.46127615962177862544729767791, −2.92536786962684725682747564756, −0.29076617494332701847985117709, 2.33592624091734995189145202453, 4.01447443399732412499039903522, 7.13654618147931156828275988231, 7.940501861850310078121225330530, 9.328004399158371282564257148063, 10.52544723917552025142171411106, 11.81914220731919902008837708814, 12.84501713531221302767639927364, 14.78178332500309035553767097568, 15.34340832392024494430835456630

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