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Properties

Label 2-40-40.37-c4-0-13
Degree $2$
Conductor $40$
Sign $0.298 - 0.954i$
Analytic cond. $4.13479$
Root an. cond. $2.03342$
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  + (3.23 + 2.35i)2-s + (9.60 + 9.60i)3-s + (4.89 + 15.2i)4-s + (12.7 − 21.4i)5-s + (8.41 + 53.6i)6-s + (−60.7 − 60.7i)7-s + (−20.0 + 60.7i)8-s + 103. i·9-s + (91.9 − 39.2i)10-s − 85.0i·11-s + (−99.2 + 193. i)12-s + (41.8 + 41.8i)13-s + (−53.2 − 339. i)14-s + (329. − 83.3i)15-s + (−208. + 149. i)16-s + (181. + 181. i)17-s + ⋯
L(s)  = 1  + (0.808 + 0.589i)2-s + (1.06 + 1.06i)3-s + (0.305 + 0.952i)4-s + (0.511 − 0.859i)5-s + (0.233 + 1.49i)6-s + (−1.24 − 1.24i)7-s + (−0.313 + 0.949i)8-s + 1.27i·9-s + (0.919 − 0.392i)10-s − 0.703i·11-s + (−0.689 + 1.34i)12-s + (0.247 + 0.247i)13-s + (−0.271 − 1.73i)14-s + (1.46 − 0.370i)15-s + (−0.812 + 0.582i)16-s + (0.629 + 0.629i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.298 - 0.954i$
Analytic conductor: \(4.13479\)
Root analytic conductor: \(2.03342\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :2),\ 0.298 - 0.954i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.27495 + 1.67247i\)
\(L(\frac12)\) \(\approx\) \(2.27495 + 1.67247i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.23 - 2.35i)T \)
5 \( 1 + (-12.7 + 21.4i)T \)
good3 \( 1 + (-9.60 - 9.60i)T + 81iT^{2} \)
7 \( 1 + (60.7 + 60.7i)T + 2.40e3iT^{2} \)
11 \( 1 + 85.0iT - 1.46e4T^{2} \)
13 \( 1 + (-41.8 - 41.8i)T + 2.85e4iT^{2} \)
17 \( 1 + (-181. - 181. i)T + 8.35e4iT^{2} \)
19 \( 1 + 260.T + 1.30e5T^{2} \)
23 \( 1 + (180. - 180. i)T - 2.79e5iT^{2} \)
29 \( 1 - 1.11e3T + 7.07e5T^{2} \)
31 \( 1 + 571.T + 9.23e5T^{2} \)
37 \( 1 + (273. - 273. i)T - 1.87e6iT^{2} \)
41 \( 1 + 503.T + 2.82e6T^{2} \)
43 \( 1 + (-2.05e3 - 2.05e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-66.4 - 66.4i)T + 4.87e6iT^{2} \)
53 \( 1 + (518. + 518. i)T + 7.89e6iT^{2} \)
59 \( 1 + 5.24e3T + 1.21e7T^{2} \)
61 \( 1 + 2.27e3iT - 1.38e7T^{2} \)
67 \( 1 + (-4.86e3 + 4.86e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 7.97e3T + 2.54e7T^{2} \)
73 \( 1 + (-293. + 293. i)T - 2.83e7iT^{2} \)
79 \( 1 - 4.82e3iT - 3.89e7T^{2} \)
83 \( 1 + (-748. - 748. i)T + 4.74e7iT^{2} \)
89 \( 1 + 6.61e3iT - 6.27e7T^{2} \)
97 \( 1 + (-7.80e3 - 7.80e3i)T + 8.85e7iT^{2} \)
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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

−15.75968030368052159481193205240, −14.28270525910365987929378931351, −13.63238950641961054448018477881, −12.66270743836889841825335348704, −10.48263146597078195619555277848, −9.292169026481901723247985066868, −8.117308061629742907689701773306, −6.23269181389469683886039575225, −4.37424018087384191574337433588, −3.33475842122793751863707936820, 2.22248206047021378231811613143, 3.09564255041428634942949790380, 5.97842252136694765849347766826, 7.03602336913692983533642181644, 9.062313386945514385164399384205, 10.21660614747358118215200428783, 12.14200138851494614613427951541, 12.83776913796617457926800720827, 13.86189544183421319659426340844, 14.75789297593883795113366954946

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