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Properties

Label 2-405-135.68-c1-0-15
Degree $2$
Conductor $405$
Sign $-0.980 - 0.194i$
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  + (1.48 − 2.12i)2-s + (−1.60 − 4.41i)4-s + (−1.13 − 1.92i)5-s + (−2.68 + 1.25i)7-s + (−6.75 − 1.80i)8-s + (−5.77 − 0.451i)10-s + (1.00 + 1.19i)11-s + (3.28 − 2.29i)13-s + (−1.33 + 7.54i)14-s + (−6.66 + 5.59i)16-s + (−1.82 + 0.490i)17-s + (2.41 − 1.39i)19-s + (−6.68 + 8.11i)20-s + (4.02 − 0.352i)22-s + (3.16 − 6.78i)23-s + ⋯
L(s)  = 1  + (1.04 − 1.49i)2-s + (−0.803 − 2.20i)4-s + (−0.507 − 0.861i)5-s + (−1.01 + 0.473i)7-s + (−2.38 − 0.639i)8-s + (−1.82 − 0.142i)10-s + (0.302 + 0.360i)11-s + (0.910 − 0.637i)13-s + (−0.355 + 2.01i)14-s + (−1.66 + 1.39i)16-s + (−0.443 + 0.118i)17-s + (0.553 − 0.319i)19-s + (−1.49 + 1.81i)20-s + (0.859 − 0.0751i)22-s + (0.659 − 1.41i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.980 - 0.194i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -0.980 - 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.170380 + 1.73126i\)
\(L(\frac12)\) \(\approx\) \(0.170380 + 1.73126i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 + 1.92i)T \)
good2 \( 1 + (-1.48 + 2.12i)T + (-0.684 - 1.87i)T^{2} \)
7 \( 1 + (2.68 - 1.25i)T + (4.49 - 5.36i)T^{2} \)
11 \( 1 + (-1.00 - 1.19i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-3.28 + 2.29i)T + (4.44 - 12.2i)T^{2} \)
17 \( 1 + (1.82 - 0.490i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.41 + 1.39i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.16 + 6.78i)T + (-14.7 - 17.6i)T^{2} \)
29 \( 1 + (0.683 + 3.87i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-5.45 + 1.98i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-0.0847 - 0.316i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.26 + 1.10i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.582 - 0.0509i)T + (42.3 + 7.46i)T^{2} \)
47 \( 1 + (0.321 + 0.690i)T + (-30.2 + 36.0i)T^{2} \)
53 \( 1 + (5.57 - 5.57i)T - 53iT^{2} \)
59 \( 1 + (-7.84 - 6.57i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-11.4 - 4.15i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.14 - 1.63i)T + (-22.9 + 62.9i)T^{2} \)
71 \( 1 + (-6.11 - 3.52i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.05 - 3.93i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-12.9 + 2.28i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-3.33 - 2.33i)T + (28.3 + 77.9i)T^{2} \)
89 \( 1 + (-4.23 - 7.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.696 + 7.95i)T + (-95.5 - 16.8i)T^{2} \)
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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

−11.05455132682455137874014426388, −10.06843697831560611489649609684, −9.274421472032927587578132776095, −8.388187319999732526084319451640, −6.57241544566175185116205080457, −5.55375562060548498552337690155, −4.54376142860549327009559312660, −3.63561838080034885480931011402, −2.56524940459032873141417096312, −0.857578839285124801100603679269, 3.37701898240466687590532272158, 3.70781623959915672977723279599, 5.11904148450439299643512488190, 6.43664040995331097812641772042, 6.68328531872735951443959922603, 7.62610691163413474577910007603, 8.618782153325927396971517117202, 9.736445853100281842854739804857, 11.10299669537791505741998652836, 11.89758494575323408849850736133

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