Properties

Label 2-40-40.13-c10-0-43
Degree $2$
Conductor $40$
Sign $0.973 + 0.230i$
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  + (−23.6 + 21.5i)2-s + (119. − 119. i)3-s + (98.9 − 1.01e3i)4-s + (2.96e3 + 981. i)5-s + (−260. + 5.38e3i)6-s + (2.01e4 − 2.01e4i)7-s + (1.95e4 + 2.62e4i)8-s + 3.06e4i·9-s + (−9.14e4 + 4.05e4i)10-s + 5.73e4i·11-s + (−1.09e5 − 1.33e5i)12-s + (3.12e5 − 3.12e5i)13-s + (−4.41e4 + 9.10e5i)14-s + (4.70e5 − 2.36e5i)15-s + (−1.02e6 − 2.01e5i)16-s + (4.98e5 − 4.98e5i)17-s + ⋯
L(s)  = 1  + (−0.740 + 0.672i)2-s + (0.490 − 0.490i)3-s + (0.0966 − 0.995i)4-s + (0.949 + 0.313i)5-s + (−0.0335 + 0.692i)6-s + (1.19 − 1.19i)7-s + (0.597 + 0.801i)8-s + 0.519i·9-s + (−0.914 + 0.405i)10-s + 0.356i·11-s + (−0.440 − 0.535i)12-s + (0.842 − 0.842i)13-s + (−0.0820 + 1.69i)14-s + (0.619 − 0.311i)15-s + (−0.981 − 0.192i)16-s + (0.351 − 0.351i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.973 + 0.230i$
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: \((2,\ 40,\ (\ :5),\ 0.973 + 0.230i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.21257 - 0.258688i\)
\(L(\frac12)\) \(\approx\) \(2.21257 - 0.258688i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (23.6 - 21.5i)T \)
5 \( 1 + (-2.96e3 - 981. i)T \)
good3 \( 1 + (-119. + 119. i)T - 5.90e4iT^{2} \)
7 \( 1 + (-2.01e4 + 2.01e4i)T - 2.82e8iT^{2} \)
11 \( 1 - 5.73e4iT - 2.59e10T^{2} \)
13 \( 1 + (-3.12e5 + 3.12e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (-4.98e5 + 4.98e5i)T - 2.01e12iT^{2} \)
19 \( 1 + 4.17e6T + 6.13e12T^{2} \)
23 \( 1 + (2.67e6 + 2.67e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 2.83e7T + 4.20e14T^{2} \)
31 \( 1 + 1.38e6T + 8.19e14T^{2} \)
37 \( 1 + (-1.65e7 - 1.65e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 4.47e7T + 1.34e16T^{2} \)
43 \( 1 + (-7.71e7 + 7.71e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (2.63e8 - 2.63e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (-4.81e8 + 4.81e8i)T - 1.74e17iT^{2} \)
59 \( 1 - 2.22e8T + 5.11e17T^{2} \)
61 \( 1 + 6.74e8iT - 7.13e17T^{2} \)
67 \( 1 + (1.08e9 + 1.08e9i)T + 1.82e18iT^{2} \)
71 \( 1 + 5.34e8T + 3.25e18T^{2} \)
73 \( 1 + (4.97e8 + 4.97e8i)T + 4.29e18iT^{2} \)
79 \( 1 + 2.37e9iT - 9.46e18T^{2} \)
83 \( 1 + (-2.94e9 + 2.94e9i)T - 1.55e19iT^{2} \)
89 \( 1 - 5.94e9iT - 3.11e19T^{2} \)
97 \( 1 + (9.87e9 - 9.87e9i)T - 7.37e19iT^{2} \)
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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

−14.08939714624735364203097190886, −13.23327571864160868238867088692, −10.83579913005006500622666774348, −10.29977088508393692106223135340, −8.513852249521941015412854364291, −7.69073690796552468993659270641, −6.44040234441496184301403212328, −4.82867041129041002910725549890, −2.13213845042434122200318165144, −1.04344904140602685159819233218, 1.39030268468716292533303933998, 2.48844252635553687421413593816, 4.28341039180858091218441156819, 6.14671830435670497917613295330, 8.579904526039532798886968080162, 8.779979481337256799281013302308, 10.14551604203771415049105553116, 11.43468002501503134294297406480, 12.53046768474448557788693879207, 13.98327605249289220034198124083

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