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Properties

Label 2-405-45.13-c2-0-9
Degree $2$
Conductor $405$
Sign $0.878 - 0.477i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.00 − 0.803i)2-s + (4.89 + 2.82i)4-s + (−4.69 + 1.73i)5-s + (−1.09 − 0.294i)7-s + (−3.62 − 3.62i)8-s + (15.4 − 1.42i)10-s + (−9.39 − 16.2i)11-s + (4.80 − 1.28i)13-s + (3.05 + 1.76i)14-s + (−3.34 − 5.79i)16-s + (−10.4 + 10.4i)17-s + 0.359i·19-s + (−27.8 − 4.77i)20-s + (15.1 + 56.3i)22-s + (27.2 − 7.29i)23-s + ⋯
L(s)  = 1  + (−1.50 − 0.401i)2-s + (1.22 + 0.706i)4-s + (−0.938 + 0.346i)5-s + (−0.156 − 0.0420i)7-s + (−0.452 − 0.452i)8-s + (1.54 − 0.142i)10-s + (−0.854 − 1.47i)11-s + (0.369 − 0.0990i)13-s + (0.218 + 0.126i)14-s + (−0.209 − 0.362i)16-s + (−0.613 + 0.613i)17-s + 0.0189i·19-s + (−1.39 − 0.238i)20-s + (0.686 + 2.56i)22-s + (1.18 − 0.316i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.878 - 0.477i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.878 - 0.477i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.406576 + 0.103424i\)
\(L(\frac12)\) \(\approx\) \(0.406576 + 0.103424i\)
\(L(2)\) 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 \( 1 + (4.69 - 1.73i)T \)
good2 \( 1 + (3.00 + 0.803i)T + (3.46 + 2i)T^{2} \)
7 \( 1 + (1.09 + 0.294i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (9.39 + 16.2i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.80 + 1.28i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (10.4 - 10.4i)T - 289iT^{2} \)
19 \( 1 - 0.359iT - 361T^{2} \)
23 \( 1 + (-27.2 + 7.29i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (43.6 - 25.1i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (13.9 - 24.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (9.24 - 9.24i)T - 1.36e3iT^{2} \)
41 \( 1 + (-25.5 + 44.1i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (4.85 - 18.1i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-83.8 - 22.4i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-52.9 - 52.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-71.6 - 41.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-41.9 - 72.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-1.04 - 3.89i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 26.1T + 5.04e3T^{2} \)
73 \( 1 + (-51.0 - 51.0i)T + 5.32e3iT^{2} \)
79 \( 1 + (-92.4 + 53.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (6.92 - 25.8i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 41.9iT - 7.92e3T^{2} \)
97 \( 1 + (48.9 + 13.1i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−10.77773780029412975603950447516, −10.57242032123594050300123124181, −8.971165324228156387579152381575, −8.623178324198475784522324819555, −7.67054418392439464816182747155, −6.87810405532027746156232359333, −5.44680921822254956157728998824, −3.72188013066211638421785711111, −2.64948541575772210175735491223, −0.825157917870953266038887541427, 0.44056215300376344279190542079, 2.15299786249788816490304105561, 4.01994468387396864416678651765, 5.22425093953677106054972119863, 6.84102623994558257318837269319, 7.44771013124275792060619774957, 8.144569881470651841636074137457, 9.201017424351847212007118541049, 9.700345476930001715087452275737, 10.87334108152810534336822520613

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