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Properties

Label 2-40-40.29-c5-0-11
Degree $2$
Conductor $40$
Sign $-0.385 - 0.922i$
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  + (−0.685 + 5.61i)2-s + 28.9·3-s + (−31.0 − 7.69i)4-s + (−40.5 + 38.5i)5-s + (−19.8 + 162. i)6-s + 128. i·7-s + (64.5 − 169. i)8-s + 592.·9-s + (−188. − 253. i)10-s + 433. i·11-s + (−897. − 222. i)12-s + 78.5·13-s + (−719. − 87.8i)14-s + (−1.17e3 + 1.11e3i)15-s + (905. + 478. i)16-s − 1.48e3i·17-s + ⋯
L(s)  = 1  + (−0.121 + 0.992i)2-s + 1.85·3-s + (−0.970 − 0.240i)4-s + (−0.724 + 0.689i)5-s + (−0.224 + 1.84i)6-s + 0.988i·7-s + (0.356 − 0.934i)8-s + 2.43·9-s + (−0.596 − 0.802i)10-s + 1.08i·11-s + (−1.79 − 0.446i)12-s + 0.128·13-s + (−0.980 − 0.119i)14-s + (−1.34 + 1.27i)15-s + (0.884 + 0.466i)16-s − 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.385 - 0.922i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ -0.385 - 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.21381 + 1.82297i\)
\(L(\frac12)\) \(\approx\) \(1.21381 + 1.82297i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.685 - 5.61i)T \)
5 \( 1 + (40.5 - 38.5i)T \)
good3 \( 1 - 28.9T + 243T^{2} \)
7 \( 1 - 128. iT - 1.68e4T^{2} \)
11 \( 1 - 433. iT - 1.61e5T^{2} \)
13 \( 1 - 78.5T + 3.71e5T^{2} \)
17 \( 1 + 1.48e3iT - 1.41e6T^{2} \)
19 \( 1 + 98.3iT - 2.47e6T^{2} \)
23 \( 1 + 2.36e3iT - 6.43e6T^{2} \)
29 \( 1 + 1.02e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.30e3T + 2.86e7T^{2} \)
37 \( 1 - 6.64e3T + 6.93e7T^{2} \)
41 \( 1 + 4.44e3T + 1.15e8T^{2} \)
43 \( 1 - 7.06e3T + 1.47e8T^{2} \)
47 \( 1 + 1.55e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.67e4T + 4.18e8T^{2} \)
59 \( 1 - 3.29e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.92e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.06e3T + 1.35e9T^{2} \)
71 \( 1 - 1.23e4T + 1.80e9T^{2} \)
73 \( 1 + 3.72e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.07e4T + 3.07e9T^{2} \)
83 \( 1 - 3.86e4T + 3.93e9T^{2} \)
89 \( 1 + 1.32e5T + 5.58e9T^{2} \)
97 \( 1 - 9.50e4iT - 8.58e9T^{2} \)
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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

−15.22943415514232198121193338855, −14.71074782833192643874307766297, −13.68766489158543850441111611127, −12.33440854237290633751636120299, −9.951703058611029060696827619691, −8.923207968867038035750980182551, −7.894334671937143909094986177872, −6.91305126090579118578533852994, −4.39687187618698848402252934508, −2.69869577149243655531011324308, 1.26272580867447112173123678550, 3.32302135982018593520332729521, 4.19514939469109799682650970221, 7.81833131841994496218190895878, 8.477586353087711642113099706960, 9.628759956477008538144128086411, 10.99221680217266260695940490752, 12.70991451050142562678566329911, 13.50935160477743369128356061179, 14.36830217566274128997337025222

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