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Properties

Label 2-40-40.13-c2-0-0
Degree $2$
Conductor $40$
Sign $-0.996 - 0.0812i$
Analytic cond. $1.08992$
Root an. cond. $1.04399$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.46 + 1.35i)2-s + (−2.57 + 2.57i)3-s + (0.312 − 3.98i)4-s + (−4.90 − 0.973i)5-s + (0.284 − 7.27i)6-s + (−4.07 + 4.07i)7-s + (4.95 + 6.27i)8-s − 4.26i·9-s + (8.52 − 5.22i)10-s + 16.0i·11-s + (9.46 + 11.0i)12-s + (9.77 − 9.77i)13-s + (0.450 − 11.5i)14-s + (15.1 − 10.1i)15-s + (−15.8 − 2.49i)16-s + (−12.0 + 12.0i)17-s + ⋯
L(s)  = 1  + (−0.734 + 0.678i)2-s + (−0.858 + 0.858i)3-s + (0.0780 − 0.996i)4-s + (−0.980 − 0.194i)5-s + (0.0474 − 1.21i)6-s + (−0.582 + 0.582i)7-s + (0.619 + 0.784i)8-s − 0.473i·9-s + (0.852 − 0.522i)10-s + 1.46i·11-s + (0.788 + 0.922i)12-s + (0.752 − 0.752i)13-s + (0.0321 − 0.822i)14-s + (1.00 − 0.674i)15-s + (−0.987 − 0.155i)16-s + (−0.709 + 0.709i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0812i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.996 - 0.0812i$
Analytic conductor: \(1.08992\)
Root analytic conductor: \(1.04399\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1),\ -0.996 - 0.0812i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0138529 + 0.340247i\)
\(L(\frac12)\) \(\approx\) \(0.0138529 + 0.340247i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.46 - 1.35i)T \)
5 \( 1 + (4.90 + 0.973i)T \)
good3 \( 1 + (2.57 - 2.57i)T - 9iT^{2} \)
7 \( 1 + (4.07 - 4.07i)T - 49iT^{2} \)
11 \( 1 - 16.0iT - 121T^{2} \)
13 \( 1 + (-9.77 + 9.77i)T - 169iT^{2} \)
17 \( 1 + (12.0 - 12.0i)T - 289iT^{2} \)
19 \( 1 - 9.55T + 361T^{2} \)
23 \( 1 + (2.56 + 2.56i)T + 529iT^{2} \)
29 \( 1 + 10.1T + 841T^{2} \)
31 \( 1 + 24.9T + 961T^{2} \)
37 \( 1 + (-26.8 - 26.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 62.0T + 1.68e3T^{2} \)
43 \( 1 + (13.5 - 13.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (-11.9 + 11.9i)T - 2.20e3iT^{2} \)
53 \( 1 + (45.8 - 45.8i)T - 2.80e3iT^{2} \)
59 \( 1 + 28.9T + 3.48e3T^{2} \)
61 \( 1 - 68.7iT - 3.72e3T^{2} \)
67 \( 1 + (-14.7 - 14.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 31.1T + 5.04e3T^{2} \)
73 \( 1 + (25.6 + 25.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 70.6iT - 6.24e3T^{2} \)
83 \( 1 + (-96.2 + 96.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 103. iT - 7.92e3T^{2} \)
97 \( 1 + (-30.8 + 30.8i)T - 9.40e3iT^{2} \)
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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

−16.30754195888342031944690511823, −15.59803020071910462734195772797, −15.01447632795257765128593625166, −12.77353246126682201979284252081, −11.36288129230975088541597946896, −10.33237837542996057158793267189, −9.127146420533648328701087381734, −7.62601897060159232794318080820, −5.98502064791576085559634503928, −4.53797841665221042304165441445, 0.53707005397648237743472913502, 3.63486106860620215838887084383, 6.49619781547466621238006604939, 7.59771386491267697550756316543, 9.093264524019060337744286885792, 11.09076496261921344588059396362, 11.37635776734974702277877452850, 12.67255891593697125022435103103, 13.72687525124763134281158835192, 16.08806239042091779749182869025

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