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Properties

Label 2-40-40.27-c5-0-4
Degree $2$
Conductor $40$
Sign $-0.267 - 0.963i$
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  + (0.0746 − 5.65i)2-s + (−5.85 + 5.85i)3-s + (−31.9 − 0.844i)4-s + (13.4 − 54.2i)5-s + (32.6 + 33.5i)6-s + (−100. + 100. i)7-s + (−7.16 + 180. i)8-s + 174. i·9-s + (−305. − 80.0i)10-s − 546.·11-s + (192. − 182. i)12-s + (619. + 619. i)13-s + (560. + 575. i)14-s + (239. + 396. i)15-s + (1.02e3 + 54.0i)16-s + (−1.50e3 − 1.50e3i)17-s + ⋯
L(s)  = 1  + (0.0131 − 0.999i)2-s + (−0.375 + 0.375i)3-s + (−0.999 − 0.0263i)4-s + (0.240 − 0.970i)5-s + (0.370 + 0.380i)6-s + (−0.775 + 0.775i)7-s + (−0.0395 + 0.999i)8-s + 0.717i·9-s + (−0.967 − 0.253i)10-s − 1.36·11-s + (0.385 − 0.365i)12-s + (1.01 + 1.01i)13-s + (0.764 + 0.785i)14-s + (0.274 + 0.455i)15-s + (0.998 + 0.0527i)16-s + (−1.26 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.267 - 0.963i$
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: \((2,\ 40,\ (\ :5/2),\ -0.267 - 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.125795 + 0.165562i\)
\(L(\frac12)\) \(\approx\) \(0.125795 + 0.165562i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0746 + 5.65i)T \)
5 \( 1 + (-13.4 + 54.2i)T \)
good3 \( 1 + (5.85 - 5.85i)T - 243iT^{2} \)
7 \( 1 + (100. - 100. i)T - 1.68e4iT^{2} \)
11 \( 1 + 546.T + 1.61e5T^{2} \)
13 \( 1 + (-619. - 619. i)T + 3.71e5iT^{2} \)
17 \( 1 + (1.50e3 + 1.50e3i)T + 1.41e6iT^{2} \)
19 \( 1 - 1.14e3iT - 2.47e6T^{2} \)
23 \( 1 + (2.13e3 + 2.13e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 1.30e3T + 2.05e7T^{2} \)
31 \( 1 - 4.69e3iT - 2.86e7T^{2} \)
37 \( 1 + (-7.59e3 + 7.59e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 8.81e3T + 1.15e8T^{2} \)
43 \( 1 + (745. - 745. i)T - 1.47e8iT^{2} \)
47 \( 1 + (26.5 - 26.5i)T - 2.29e8iT^{2} \)
53 \( 1 + (189. + 189. i)T + 4.18e8iT^{2} \)
59 \( 1 - 3.91e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.75e3iT - 8.44e8T^{2} \)
67 \( 1 + (-2.08e4 - 2.08e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 1.50e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.61e3 - 2.61e3i)T - 2.07e9iT^{2} \)
79 \( 1 + 1.74e4T + 3.07e9T^{2} \)
83 \( 1 + (-3.11e4 + 3.11e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 3.34e4iT - 5.58e9T^{2} \)
97 \( 1 + (-3.26e3 - 3.26e3i)T + 8.58e9iT^{2} \)
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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

−15.92820081230973684766907962704, −13.75831828873988292437792747949, −13.02391764228261663270674759810, −11.87448142761297477114391197126, −10.66486129214998932756306180039, −9.453391007540139785731627263424, −8.431071078125209867244163826630, −5.68713193238753146460079090399, −4.49258558462535281794156226241, −2.28434710608780850286518403480, 0.11611324377213165278899693912, 3.59080374040165310571856126923, 5.90579785243038293921626833490, 6.72745919781467057680433513680, 8.017760865930507991238596377603, 9.838573944290588640141649024966, 10.90983075517098897689889745630, 13.11029014210579252781162405754, 13.38160284312569348570902799969, 15.16142067233829987683753367950

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