Properties

Label 2-40-40.13-c10-0-27
Degree $2$
Conductor $40$
Sign $0.794 + 0.607i$
Analytic cond. $25.4142$
Root an. cond. $5.04125$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−24.3 − 20.7i)2-s + (279. − 279. i)3-s + (166. + 1.01e3i)4-s + (−2.82e3 + 1.33e3i)5-s + (−1.26e4 + 1.03e3i)6-s + (−1.94e4 + 1.94e4i)7-s + (1.68e4 − 2.80e4i)8-s − 9.77e4i·9-s + (9.65e4 + 2.58e4i)10-s + 1.25e5i·11-s + (3.29e5 + 2.36e5i)12-s + (3.93e5 − 3.93e5i)13-s + (8.75e5 − 7.14e4i)14-s + (−4.16e5 + 1.16e6i)15-s + (−9.93e5 + 3.35e5i)16-s + (8.53e5 − 8.53e5i)17-s + ⋯
L(s)  = 1  + (−0.762 − 0.647i)2-s + (1.15 − 1.15i)3-s + (0.162 + 0.986i)4-s + (−0.903 + 0.428i)5-s + (−1.62 + 0.132i)6-s + (−1.15 + 1.15i)7-s + (0.515 − 0.857i)8-s − 1.65i·9-s + (0.965 + 0.258i)10-s + 0.782i·11-s + (1.32 + 0.950i)12-s + (1.05 − 1.05i)13-s + (1.62 − 0.132i)14-s + (−0.548 + 1.53i)15-s + (−0.947 + 0.320i)16-s + (0.601 − 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(25.4142\)
Root analytic conductor: \(5.04125\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5),\ 0.794 + 0.607i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.39428 - 0.471578i\)
\(L(\frac12)\) \(\approx\) \(1.39428 - 0.471578i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (24.3 + 20.7i)T \)
5 \( 1 + (2.82e3 - 1.33e3i)T \)
good3 \( 1 + (-279. + 279. i)T - 5.90e4iT^{2} \)
7 \( 1 + (1.94e4 - 1.94e4i)T - 2.82e8iT^{2} \)
11 \( 1 - 1.25e5iT - 2.59e10T^{2} \)
13 \( 1 + (-3.93e5 + 3.93e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (-8.53e5 + 8.53e5i)T - 2.01e12iT^{2} \)
19 \( 1 + 1.27e5T + 6.13e12T^{2} \)
23 \( 1 + (-6.80e6 - 6.80e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 2.84e6T + 4.20e14T^{2} \)
31 \( 1 - 4.46e7T + 8.19e14T^{2} \)
37 \( 1 + (-5.85e7 - 5.85e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 8.39e7T + 1.34e16T^{2} \)
43 \( 1 + (-4.40e7 + 4.40e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (2.21e8 - 2.21e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (7.67e7 - 7.67e7i)T - 1.74e17iT^{2} \)
59 \( 1 - 6.62e8T + 5.11e17T^{2} \)
61 \( 1 - 2.66e8iT - 7.13e17T^{2} \)
67 \( 1 + (-1.12e9 - 1.12e9i)T + 1.82e18iT^{2} \)
71 \( 1 + 1.41e9T + 3.25e18T^{2} \)
73 \( 1 + (-1.40e8 - 1.40e8i)T + 4.29e18iT^{2} \)
79 \( 1 - 1.53e9iT - 9.46e18T^{2} \)
83 \( 1 + (-1.52e9 + 1.52e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 4.22e8iT - 3.11e19T^{2} \)
97 \( 1 + (2.74e9 - 2.74e9i)T - 7.37e19iT^{2} \)
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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

−13.33293629618782557283414661651, −12.59210451352116152287493362876, −11.66338133641142247958351814912, −9.837986542657246975890171639037, −8.664252715718517745664168244936, −7.77416233144435803902305747051, −6.65788757320602688436934241101, −3.26634494011499608338172228586, −2.73593099406269125528379298789, −0.991312608675884688587092199817, 0.74153150425195752523128981734, 3.36467603176628157919460891044, 4.39120432791630592859452010307, 6.62244105456506053240904864462, 8.155123907957930533302695175034, 8.928592818216116169171994493694, 10.04005579535152497465343211685, 11.04276654903243091260944935428, 13.37183931675762703991249554825, 14.36476838374386876690749789840

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