Properties

Label 2-405-135.92-c1-0-4
Degree $2$
Conductor $405$
Sign $-0.0316 - 0.999i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 0.119i)2-s + (−0.116 + 0.0204i)4-s + (0.119 + 2.23i)5-s + (−2.68 + 1.87i)7-s + (−2.80 + 0.752i)8-s + (0.430 + 3.03i)10-s + (1.85 + 5.10i)11-s + (0.215 − 2.45i)13-s + (−3.44 + 2.88i)14-s + (−3.52 + 1.28i)16-s + (3.45 + 0.925i)17-s + (0.417 + 0.240i)19-s + (−0.0595 − 0.256i)20-s + (3.14 + 6.75i)22-s + (4.01 − 5.73i)23-s + ⋯
L(s)  = 1  + (0.966 − 0.0845i)2-s + (−0.0580 + 0.0102i)4-s + (0.0533 + 0.998i)5-s + (−1.01 + 0.710i)7-s + (−0.992 + 0.265i)8-s + (0.136 + 0.960i)10-s + (0.560 + 1.53i)11-s + (0.0596 − 0.682i)13-s + (−0.920 + 0.772i)14-s + (−0.881 + 0.320i)16-s + (0.837 + 0.224i)17-s + (0.0957 + 0.0552i)19-s + (−0.0133 − 0.0574i)20-s + (0.671 + 1.43i)22-s + (0.836 − 1.19i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0316 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0316 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.0316 - 0.999i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -0.0316 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12393 + 1.16010i\)
\(L(\frac12)\) \(\approx\) \(1.12393 + 1.16010i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.119 - 2.23i)T \)
good2 \( 1 + (-1.36 + 0.119i)T + (1.96 - 0.347i)T^{2} \)
7 \( 1 + (2.68 - 1.87i)T + (2.39 - 6.57i)T^{2} \)
11 \( 1 + (-1.85 - 5.10i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-0.215 + 2.45i)T + (-12.8 - 2.25i)T^{2} \)
17 \( 1 + (-3.45 - 0.925i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.417 - 0.240i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.01 + 5.73i)T + (-7.86 - 21.6i)T^{2} \)
29 \( 1 + (0.0993 + 0.0833i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.509 - 2.88i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-1.67 + 6.26i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.215 + 0.256i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.45 - 5.27i)T + (-27.6 - 32.9i)T^{2} \)
47 \( 1 + (-6.80 - 9.71i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (-0.167 - 0.167i)T + 53iT^{2} \)
59 \( 1 + (-3.62 - 1.32i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.855 - 4.85i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-4.25 - 0.372i)T + (65.9 + 11.6i)T^{2} \)
71 \( 1 + (7.10 - 4.10i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.51 + 13.1i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.44 + 7.68i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.00128 + 0.0147i)T + (-81.7 + 14.4i)T^{2} \)
89 \( 1 + (-2.55 + 4.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.77 - 1.75i)T + (62.3 + 74.3i)T^{2} \)
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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

−11.84629581198131580448529680350, −10.57169338685208125893003709310, −9.732638010266989990693150939137, −8.959297187703191317367847971762, −7.51199804698055596730578531391, −6.49717143264732401954204284543, −5.77640924719070583254153610036, −4.54698607096700825346911697735, −3.37354403552302781057840167144, −2.55913681700033657167755101910, 0.808356813514134886802002065448, 3.33150518012964986064903342424, 3.94534828288221950468432048663, 5.21019896235707241373999936652, 5.97989903907316855530613732670, 6.97302105237956893252642777138, 8.428928401775509291085978175362, 9.238968747704411594855651977558, 9.910320742238438589025859272371, 11.39565941741432138277106657643

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