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Properties

Label 2-40-8.5-c5-0-4
Degree $2$
Conductor $40$
Sign $0.198 - 0.980i$
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  + (−5.64 − 0.375i)2-s + 6.67i·3-s + (31.7 + 4.24i)4-s − 25i·5-s + (2.50 − 37.6i)6-s − 38.2·7-s + (−177. − 35.8i)8-s + 198.·9-s + (−9.39 + 141. i)10-s + 491. i·11-s + (−28.3 + 211. i)12-s + 956. i·13-s + (216. + 14.3i)14-s + 166.·15-s + (988. + 269. i)16-s + 339.·17-s + ⋯
L(s)  = 1  + (−0.997 − 0.0664i)2-s + 0.428i·3-s + (0.991 + 0.132i)4-s − 0.447i·5-s + (0.0284 − 0.427i)6-s − 0.295·7-s + (−0.980 − 0.198i)8-s + 0.816·9-s + (−0.0297 + 0.446i)10-s + 1.22i·11-s + (−0.0567 + 0.424i)12-s + 1.56i·13-s + (0.294 + 0.0196i)14-s + 0.191·15-s + (0.964 + 0.262i)16-s + 0.285·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.198 - 0.980i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ 0.198 - 0.980i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.717132 + 0.586692i\)
\(L(\frac12)\) \(\approx\) \(0.717132 + 0.586692i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.64 + 0.375i)T \)
5 \( 1 + 25iT \)
good3 \( 1 - 6.67iT - 243T^{2} \)
7 \( 1 + 38.2T + 1.68e4T^{2} \)
11 \( 1 - 491. iT - 1.61e5T^{2} \)
13 \( 1 - 956. iT - 3.71e5T^{2} \)
17 \( 1 - 339.T + 1.41e6T^{2} \)
19 \( 1 - 1.86e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.99e3T + 6.43e6T^{2} \)
29 \( 1 + 3.57e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.71e3T + 2.86e7T^{2} \)
37 \( 1 + 2.83e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.06e4T + 1.15e8T^{2} \)
43 \( 1 - 2.05e4iT - 1.47e8T^{2} \)
47 \( 1 + 756.T + 2.29e8T^{2} \)
53 \( 1 + 3.16e4iT - 4.18e8T^{2} \)
59 \( 1 + 4.91e3iT - 7.14e8T^{2} \)
61 \( 1 + 2.14e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.81e3iT - 1.35e9T^{2} \)
71 \( 1 + 1.11e4T + 1.80e9T^{2} \)
73 \( 1 - 7.30e3T + 2.07e9T^{2} \)
79 \( 1 + 2.35e4T + 3.07e9T^{2} \)
83 \( 1 + 2.37e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.25e5T + 5.58e9T^{2} \)
97 \( 1 - 1.26e5T + 8.58e9T^{2} \)
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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

−15.84519958761876501616478221562, −14.53785842274043576397067899373, −12.67737891191981896283343428965, −11.69120775255059385106880181992, −9.994710979703212848855805754661, −9.525891205503952200429809090110, −7.895712072546130582042593192293, −6.52283728253734944054911109130, −4.26083480944704716504144764975, −1.73691780736693295545467658513, 0.74602661335797141341683292836, 2.94121891379844951877156945477, 5.98249485917844656774550404394, 7.27419786490068329424566102908, 8.432274728450694359380114303006, 9.999562939268215798066829196476, 10.93470722363911293768441326920, 12.35295292110439244696081021057, 13.65708321569503491263501774116, 15.30378512499379434776999826772

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