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Properties

Label 2-40-40.27-c5-0-10
Degree $2$
Conductor $40$
Sign $0.757 - 0.653i$
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  + (−5.57 − 0.949i)2-s + (−3.12 + 3.12i)3-s + (30.1 + 10.5i)4-s + (−34.4 − 44.0i)5-s + (20.4 − 14.4i)6-s + (−19.1 + 19.1i)7-s + (−158. − 87.6i)8-s + 223. i·9-s + (150. + 278. i)10-s + 648.·11-s + (−127. + 61.3i)12-s + (392. + 392. i)13-s + (125. − 88.8i)14-s + (245. + 29.9i)15-s + (799. + 639. i)16-s + (739. + 739. i)17-s + ⋯
L(s)  = 1  + (−0.985 − 0.167i)2-s + (−0.200 + 0.200i)3-s + (0.943 + 0.330i)4-s + (−0.616 − 0.787i)5-s + (0.231 − 0.164i)6-s + (−0.148 + 0.148i)7-s + (−0.874 − 0.484i)8-s + 0.919i·9-s + (0.475 + 0.879i)10-s + 1.61·11-s + (−0.255 + 0.122i)12-s + (0.643 + 0.643i)13-s + (0.170 − 0.121i)14-s + (0.281 + 0.0343i)15-s + (0.781 + 0.624i)16-s + (0.620 + 0.620i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.757 - 0.653i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ 0.757 - 0.653i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.813533 + 0.302498i\)
\(L(\frac12)\) \(\approx\) \(0.813533 + 0.302498i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.57 + 0.949i)T \)
5 \( 1 + (34.4 + 44.0i)T \)
good3 \( 1 + (3.12 - 3.12i)T - 243iT^{2} \)
7 \( 1 + (19.1 - 19.1i)T - 1.68e4iT^{2} \)
11 \( 1 - 648.T + 1.61e5T^{2} \)
13 \( 1 + (-392. - 392. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-739. - 739. i)T + 1.41e6iT^{2} \)
19 \( 1 - 1.80e3iT - 2.47e6T^{2} \)
23 \( 1 + (1.66e3 + 1.66e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 7.68e3T + 2.05e7T^{2} \)
31 \( 1 + 2.50e3iT - 2.86e7T^{2} \)
37 \( 1 + (4.61e3 - 4.61e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 1.66e3T + 1.15e8T^{2} \)
43 \( 1 + (-1.13e4 + 1.13e4i)T - 1.47e8iT^{2} \)
47 \( 1 + (1.29e4 - 1.29e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-1.26e4 - 1.26e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 3.00e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.27e4iT - 8.44e8T^{2} \)
67 \( 1 + (6.33e3 + 6.33e3i)T + 1.35e9iT^{2} \)
71 \( 1 - 2.95e3iT - 1.80e9T^{2} \)
73 \( 1 + (1.07e4 - 1.07e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 7.47e3T + 3.07e9T^{2} \)
83 \( 1 + (-9.87e3 + 9.87e3i)T - 3.93e9iT^{2} \)
89 \( 1 + 226. iT - 5.58e9T^{2} \)
97 \( 1 + (-6.96e4 - 6.96e4i)T + 8.58e9iT^{2} \)
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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

−15.82869496562633045098950437100, −14.23627212105142625992952521613, −12.37451667804469027584040767601, −11.63603413562960129430291326003, −10.28510608693862358071025968011, −8.931634179102444113391060374945, −7.977994102297388108628062191016, −6.24451933632882220728981634009, −4.01558135434317862566894041709, −1.39217848406253807815776320454, 0.802084154790235834106749389771, 3.37424178571947426989179313079, 6.30771623025786184337409645894, 7.15585602321047167161515951574, 8.717772836908787603927157415919, 9.985122359008330711159200310902, 11.37871433398974317127957289431, 12.06755452089938950236080615646, 14.19904730396606594014977089736, 15.23734101974435418591721811137

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