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Properties

Label 2-405-15.14-c4-0-21
Degree $2$
Conductor $405$
Sign $0.593 - 0.804i$
Analytic cond. $41.8648$
Root an. cond. $6.47030$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.50·2-s + 40.3·4-s + (−20.1 − 14.8i)5-s + 69.8i·7-s − 182.·8-s + (151. + 111. i)10-s − 52.1i·11-s − 120. i·13-s − 524. i·14-s + 727.·16-s + 80.3·17-s − 254.·19-s + (−812. − 598. i)20-s + 391. i·22-s + 250.·23-s + ⋯
L(s)  = 1  − 1.87·2-s + 2.52·4-s + (−0.804 − 0.593i)5-s + 1.42i·7-s − 2.85·8-s + (1.51 + 1.11i)10-s − 0.430i·11-s − 0.712i·13-s − 2.67i·14-s + 2.84·16-s + 0.278·17-s − 0.704·19-s + (−2.03 − 1.49i)20-s + 0.808i·22-s + 0.473·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.593 - 0.804i$
Analytic conductor: \(41.8648\)
Root analytic conductor: \(6.47030\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :2),\ 0.593 - 0.804i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4444856016\)
\(L(\frac12)\) \(\approx\) \(0.4444856016\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (20.1 + 14.8i)T \)
good2 \( 1 + 7.50T + 16T^{2} \)
7 \( 1 - 69.8iT - 2.40e3T^{2} \)
11 \( 1 + 52.1iT - 1.46e4T^{2} \)
13 \( 1 + 120. iT - 2.85e4T^{2} \)
17 \( 1 - 80.3T + 8.35e4T^{2} \)
19 \( 1 + 254.T + 1.30e5T^{2} \)
23 \( 1 - 250.T + 2.79e5T^{2} \)
29 \( 1 - 289. iT - 7.07e5T^{2} \)
31 \( 1 - 22.1T + 9.23e5T^{2} \)
37 \( 1 + 2.47e3iT - 1.87e6T^{2} \)
41 \( 1 + 486. iT - 2.82e6T^{2} \)
43 \( 1 - 99.8iT - 3.41e6T^{2} \)
47 \( 1 + 3.47e3T + 4.87e6T^{2} \)
53 \( 1 + 2.91e3T + 7.89e6T^{2} \)
59 \( 1 - 1.75e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.58e3T + 1.38e7T^{2} \)
67 \( 1 + 3.84e3iT - 2.01e7T^{2} \)
71 \( 1 + 988. iT - 2.54e7T^{2} \)
73 \( 1 + 6.01e3iT - 2.83e7T^{2} \)
79 \( 1 - 63.2T + 3.89e7T^{2} \)
83 \( 1 + 1.45e3T + 4.74e7T^{2} \)
89 \( 1 - 1.28e4iT - 6.27e7T^{2} \)
97 \( 1 - 7.32e3iT - 8.85e7T^{2} \)
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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

−10.72289377468519606859865440775, −9.530598118574962151273060625394, −8.836682295870008696637815095012, −8.298486467996181409348247437940, −7.52541149552593904953153751872, −6.30522860043496469333633773227, −5.27885590285950034539028276227, −3.27475506500127641556347542488, −2.07899088698587323432231014624, −0.70059281624242254504351232532, 0.37344558923294419530032071790, 1.58522501703472455580947225132, 3.07476569650784089068978631700, 4.35887329371167764530038428610, 6.57599574628217716175131516477, 6.98047252344949746630674141445, 7.84136662978834289195290881512, 8.518209633012345331288500241702, 9.829112309996090310686053119005, 10.22979556481406923678271518215

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